Series MI-01 · 682 instruments catalogued
Mathematics
Geometry, algebra, sequences and number theory — worked, not just answered.
In Service
682 built and verified(Reduced) Row Echelon Form Calculator
Eliminate a variable from a 2-equation system's second row, the core step row reduction repeats to reach row echelon form.
In service MI-01-0032D Distance Calculator
Find the straight-line distance between two coordinates on a plane: d = √((x₂−x₁)² + (y₂−y₁)²), built from the Pythagorean theorem.
In service MI-01-0043 Sides Triangle Area Calculator
Enter three side lengths and get the enclosed area via Heron's formula, semi-perimeter shown at every step — no angle or height needed.
In service MI-01-00530 60 90 Triangle Calculator
Enter the short leg of a 30-60-90 triangle and get the hypotenuse and long leg from the fixed 1 : √3 : 2 ratio, no Pythagorean solving needed.
In service MI-01-00645 45 90 Triangle Calculator
Enter one leg of an isosceles right triangle and get the hypotenuse instantly: hyp = leg × √2, the fixed 1:1:√2 ratio, worked step by step.
In service MI-01-007a+bi Form Calculator
Convert a complex number's magnitude and angle into rectangular a + bi form, using a = r·cos(θ) and b = r·sin(θ), with every step of the conversion shown.
In service MI-01-008AAA Triangle Calculator
Test whether three given angles could actually form a triangle — they must sum to exactly 180°, the one hard requirement AAA checks.
In service MI-01-009AAS Triangle Calculator
Enter two angles and the side opposite one of them to solve the third angle and both remaining sides with the Law of Sines.
In service MI-01-010ABC Triangle Calculator
Enter side lengths a, b, and c and recover every angle by the Law of Cosines — no angle needed to start, every step of the working shown.
In service MI-01-011Absolute Change Calculator
Find how far a value moved between two readings with direction stripped out — change = |new − old|, worked to the exact figure.
In service MI-01-012Absolute Value Calculator
Enter any real number and get its absolute value instantly, with the sign-drop rule and the distance-from-zero reasoning shown in full.
In service MI-01-013Absolute Value Equation Calculator
Solve |x − a| = b for the two points that sit exactly b units from a on the number line, one above and one below, with both cases shown.
In service MI-01-014Absolute Value Inequalities Calculator
Enter a center a and a distance b to solve |x − a| < b for its open interval — lower = a − b and upper = a + b, worked out in full.
In service MI-01-015Acute Triangle Calculator
Feed in three side lengths and find out whether the triangle they form is acute, right, or obtuse — no protractor needed.
In service MI-01-016Adding and Subtracting Fractions Calculator
Cross-multiply two fractions to a shared denominator once, then read off both their sum and their difference from the same pair of products.
In service MI-01-017Adding and Subtracting Polynomials Calculator
Add or subtract two quadratics coefficient by coefficient — x² with x², x with x, constant with constant — sum and difference shown together.
In service MI-01-018Adding Fractions Calculator
Add two fractions with different denominators using cross-multiplication, a/b + c/d = (ad + bc) / bd, with every step shown plainly.
In service MI-01-019Addition Calculator
Add two numbers — whole, negative, or decimal — and get their exact sum, the single operation every calculation on this site ultimately rests on.
In service MI-01-020Adjoint Matrix Calculator
The adjoint (classical adjugate) of a 2×2 matrix swaps the diagonal and flips the sign of the off-diagonal — the term behind every hand-computed matrix inverse.
In service MI-01-021AND Calculator
Multiply two 0-or-1 inputs to get their logical AND — true only when both are true, worked through the full truth table.
In service MI-01-022Angle Between Two Vectors Calculator
Find the angle enclosed between two 2D vectors from their x and y components — dot product, magnitudes, and the inverse cosine, worked in full.
In service MI-01-023Angle of Right Triangle Calculator
Enter a right triangle's opposite and adjacent legs and get the angle between them from θ = atan(opposite ⁄ adjacent) — no hypotenuse required.
In service MI-01-024Antilog Calculator – Antilogarithm
Enter a logarithm value and a base to recover the original number: antilog is simply raising the base to that power, undone step by step.
In service MI-01-025Arc Length Calculator
Turn a radius and a central angle into the length of the curve it sweeps. Enter degrees, radians, or turns and s = rθ resolves once the angle is converted.
In service MI-01-026Arccos Calculator (Inverse Cosine)
Enter any x between −1 and 1 to get arccos(x), the angle whose cosine is x, shown in degrees, radians, or turns, with the identity behind it explained.
In service MI-01-027Arcsin Calculator (Inverse Sine)
Recover the angle whose sine is x, restricted to arcsin's principal range of −90° to 90°, with results in degrees, radians, or turns.
In service MI-01-028Arctan Calculator (Inverse Tangent)
Find the angle whose tangent is x — arctan accepts any real number and returns the principal angle in degrees, radians, or turns.
In service MI-01-029Arcus Tangent Calculator
Recover the true direction of a point (x, y) from its y and x components separately — atan2 places the angle in the correct quadrant, all 360° of it.
In service MI-01-030Area Calculator
Pick a basic shape and get its area instantly: circle, square, rectangle, or triangle, all from one page with a shape selector.
In service MI-01-031Area of a Circle Calculator
Enter a radius and get a circle's exact area from A = πr², plus the ring-unrolling proof that shows why the radius is squared.
In service MI-01-032Area of a Frustum of a Cone Calculator
Find the curved side area and the full outer surface of a cone with its tip sliced off, from a bottom radius, a top radius, and a slant height.
In service MI-01-033Area of a Rectangle Calculator
Multiply two side lengths to get a rectangle's area instantly — enter Length and Width, read Area, with the unit-square reasoning shown.
In service MI-01-034Area of a Regular Polygon Calculator
Find the area of any regular polygon from its side count and one side length — A = ns² ⁄ (4·tan(π ⁄ n)), valid for a triangle, a 40-gon, or beyond.
In service MI-01-035Area of a Right Triangle Calculator
Multiply a right triangle's two legs together and halve it — A = ½ × base × height, using each leg directly as base or height, no altitude to construct.
In service MI-01-036Area of a Sphere Calculator
Turn a sphere's radius into its curved surface area, plus the Archimedes result that explains why the constant multiplying it is exactly four.
In service MI-01-037Area of a Square Calculator
Enter one side and get the area of a square instantly, with the squaring rule shown step by step — A = s², not 4s.
In service MI-01-038Area of a Trapezoid Calculator
Enter the two parallel bases and the height to get a trapezoid's area from A = (a + b) ⁄ 2 × h, with the midsegment shortcut explained.
In service MI-01-039Area of a Triangle SAS Calculator
Enter two sides and the angle trapped between them to get the triangle's area — no height measurement, no third side, just A = ½ab·sin(C).
In service MI-01-040Area of an Oblique Triangle Calculator
Compute a triangle's area from one side and its two adjacent angles using the Law of Sines, without measuring a second side or all three.
In service MI-01-041Area of an Obtuse Triangle Calculator
Find an obtuse triangle's area from its base and perpendicular height — the same half-base-times-height formula, its foot falling outside the base.
In service MI-01-042Area of Crescent Calculator
Find the crescent-shaped area left when a smaller circle sits entirely inside a larger one: A = π(R² − r²).
In service MI-01-043Area of Quadrilateral Calculator
Enter both diagonals and the angle where they cross to get a quadrilateral's area — A = ½·d₁·d₂·sinθ, with no side lengths required.
In service MI-01-044Area of Triangle with Coordinates Calculator
Enter three vertex coordinates and read off the enclosed area directly — the shoelace formula, with no side lengths or height to measure first.
In service MI-01-045Arithmetic Sequence Calculator
Enter a first term, common difference, and term number to get the nth term and the running sum of all terms up to that point.
In service MI-01-046ASA Triangle Calculator
Enter two angles and the side trapped between them, and this sheet returns the third angle and both remaining sides with the Law of Sines.
In service MI-01-047Associative Property Calculator
Group three numbers as (a + b) + c or as a + (b + c) and see both totals agree, with the difference between them fixed at zero.
In service MI-01-048Average Calculator
Type a list of numbers and read the arithmetic mean, the sum, and the count — formula shown, every step of the division written out.
In service MI-01-049Average Percentage Calculator
Combine two percentages correctly by weighting each one with its own sample size, not by averaging the two rates directly.
In service MI-01-050Average Rate of Change Calculator
Enter two readings of a function, x₁, f(x₁), x₂, f(x₂), and get the secant slope between them — the average rate of change, with the working shown.
In service MI-01-051Babylonian Numbers Converter
Convert any whole number from 0 to 3,599 into the ancient Babylonian base-60 system: two sexagesimal places, high and low.
In service MI-01-052Base of a Triangle Calculator
Recover a triangle's base from its area and perpendicular height — b = 2A ⁄ h, the direct rearrangement of A = ½bh, worked step by step.
In service MI-01-053Bessel Function Calculator
Approximate the Bessel function J0 with a 4-term Maclaurin series — the drumhead-vibration function with no closed form, estimated honestly for small inputs.
In service MI-01-054Bilinear Interpolation Calculator
Estimate a value at a point inside a rectangle from four known corner readings, blended by how close that point sits to each corner.
In service MI-01-055Binary Addition Calculator
Add two 4-bit binary numbers and see exactly when the result overflows past 15, wrapping around with a carry flag.
In service MI-01-056Binary Calculator
Convert any whole number from 0 to 15 into its 4-bit binary representation, one place value at a time.
In service MI-01-057Binary Division Calculator
Divide two 4-bit binary numbers and see both the quotient and the remainder broken down into their own binary digits.
In service MI-01-058Binary Fraction Converter
Convert a binary fraction, the digits after the point, into its equivalent decimal value using halving place values.
In service MI-01-059Binary Multiplication Calculator
Multiply two small binary numbers and see the product broken back down into its own four bits, with no overflow to worry about.
In service MI-01-060Binary Subtraction Calculator
Subtract one 4-bit binary number from a larger one and see the result broken back down into its own four bits.
In service MI-01-061Binomial Coefficient Calculator
Counts how many r-item subsets fit inside a set of n items using C(n,r) = n! ⁄ (r!(n−r)!), with the Pascal's-triangle addition shown alongside.
In service MI-01-062Bit Shift Calculator
Shift any number left or right by a chosen number of bits — each shift doubles or halves the value, dropping any remainder.
In service MI-01-063Bitwise Calculator
Combine two 4-bit numbers position by position using bitwise AND and OR, computed with this site's own truth-table identities.
In service MI-01-064Box Method Calculator
Split two factors into parts, fill a 2×2 grid with the four partial products, and add them back to the exact multiplication answer.
In service MI-01-065Catenary Curve Calculator
Enter a chain's tension-to-weight parameter and a horizontal distance to find how high a freely hanging cable sags: y = a·cosh(x ⁄ a), exactly.
In service MI-01-066Ceiling Function Calculator
Enter any real number and get its ceiling — the smallest integer no less than it — worked through with the rounding rule shown in full.
In service MI-01-067Center of a Circle Calculator
Recover a circle's center and radius from its expanded equation x² + y² + Dx + Ey + F = 0 by completing the square on each variable.
In service MI-01-068Center of Ellipse Calculator
Average two opposite vertices of an ellipse's major axis to find its center — the coordinate midpoint, worked with your own numbers.
In service MI-01-069Center of Mass Calculator
Enter two masses and their positions on a line to find where the pair balances — the weighted average pulled toward whichever mass is heavier.
In service MI-01-070Central Angle Calculator
Turn an arc length and radius into the central angle they subtend, reported in degrees, radians, or turns — θ = s ⁄ r worked in full.
In service MI-01-071Centroid Calculator
Find a triangle's centroid, its geometric balance point, by averaging its three vertices' x-coordinates and, separately, their y-coordinates.
In service MI-01-072Centroid of a Triangle Calculator
Enter three vertices and get the centroid — the point where a triangle balances — found by averaging the x-coordinates and the y-coordinates separately.
In service MI-01-073Change of Base Formula Calculator
Rewrite a logarithm in any base as one division of natural logs — enter the number and the base you're measuring in, and the exact answer comes back.
In service MI-01-074Characteristic Polynomial Calculator
Enter a 2×2 matrix's four entries and get its trace and determinant — the two numbers that fix the characteristic polynomial and its eigenvalues.
In service MI-01-075Check Similarity in Right Triangles Calculator
Test whether two right triangles are similar by comparing their leg ratios — matching ratios mean one triangle is a uniformly scaled copy of the other.
In service MI-01-076Chinese Remainder Theorem Calculator
Solve two remainder conditions at once: enter r1 mod m1 and r2 mod m2, and get the single x, mod their product, that satisfies both together.
In service MI-01-077Cholesky Decomposition Calculator
Enter a symmetric 2×2 matrix's three independent entries and get its lower-triangular Cholesky factor L, with the positive-definite check shown.
In service MI-01-078Chord Length Calculator
Enter a circle's radius and the angle between two radii to find the straight chord joining them: chord = 2r·sin(θ ⁄ 2), worked step by step.
In service MI-01-079Circle Calculator
Enter a circle's radius and get its area, circumference, and diameter all at once — every basic circle measure determined by that one number.
In service MI-01-080Circle Formula Calculator
Measure straight across a circle instead of to its center — enter the diameter and get area and circumference at once, from A = π(d⁄2)² and C = πd.
In service MI-01-081Circle Length Calculator
Find the length of a circular arc from a radius and a central angle — arc length = r × θ, the slice of a circle's rim that angle sweeps out.
In service MI-01-082Circle Measurements Calculator
Start from a circle's area and work backward to its radius, diameter, and circumference — r = √(A ⁄ π), solved instantly from one known figure.
In service MI-01-083Circle Perimeter Calculator
Enter a circle's radius to get its perimeter — the distance around the curve, properly called circumference — from C = 2πr.
In service MI-01-084Circle Theorems Calculator
Find the central angle corresponding to a given inscribed angle on the same arc: central = 2 × inscribed, one of geometry's oldest circle theorems.
In service MI-01-085Circumcenter Calculator
Enter a triangle's three vertices and get its circumcenter — the single point equidistant from all three corners, center of the circle through them.
In service MI-01-086Circumference and Area of a Circle Calculator
Start from a measured distance around a circle and solve backward for its radius, diameter, and area — r = C ⁄ 2π, unpacked from one taped figure.
In service MI-01-087Circumference Calculator
Turn a radius into circumference and diameter, or measure the distance around and work back to the radius — C = 2πr with every step shown.
In service MI-01-088Circumference of a Cylinder Calculator
Find how far a strap or label wraps once around a cylinder's circular cross-section, computed directly from the radius as C = 2πr.
In service MI-01-089Circumference to Diameter Calculator
Divide a measured circumference by π to get the diameter directly — no radius step, no doubling, full working shown.
In service MI-01-090Circumference-to-Radius Calculator
Divide a measured circumference by 2π to recover the exact radius, the figure that area, sector, and arc-length formulas actually call for.
In service MI-01-091Circumscribed Circle Calculator
Enter a regular polygon's side length and number of sides to find the circumradius — the radius of the one circle through every corner.
In service MI-01-092Classifying Triangles Calculator
Feed a triangle's three side lengths in and read back both its angle family and its side family from one pair of tests, in a single pass.
In service MI-01-093Clock Angle Calculator
Find the exact angle between a clock's hour and minute hands from any time, with the reflex-angle correction worked out for you.
In service MI-01-094Cofactor Expansion Calculator
Expands a 2×2 matrix along its first row into two signed terms, term1 and term2, the same row-by-row process that scales to 3×3 matrices and beyond.
In service MI-01-095Cofactor Matrix Calculator
Build the cofactor matrix of a 2×2 — each entry's signed minor, left in place, distinct from the adjugate matrix it feeds into.
In service MI-01-096Cofunction Calculator
Enter an angle to see its complement and confirm sin θ equals cos of that complement — the identity behind the name cosine.
In service MI-01-097Coin Rotation Paradox
Roll one circle around a fixed circle and count its full spins: rotations = 1 + R ⁄ r, worked step by step — not just the naive R ⁄ r guess.
In service MI-01-098Collatz Conjecture Calculator
Compute a single step of the famous Collatz sequence: halve an even number, or triple and add one to an odd one, and see why the full chain stays unsolved.
In service MI-01-099Column Space Calculator
Find the single spanning direction of a rank-deficient 2×2 matrix's column space, reduced to its simplest whole-number form.
In service MI-01-100Common Denominator Calculator
Find the least common denominator of two fractions' bottoms — the smallest number both divide into evenly, with the factoring shown.
In service MI-01-101Common Factor Calculator
Find the greatest common factor of two whole numbers with the Euclidean algorithm, then use it to reduce a fraction to lowest terms.
In service MI-01-102Common Multiple Calculator
Find the least common multiple of two whole numbers — the smallest number both divide into evenly — with the product-over-GCD identity shown.
In service MI-01-103Comparing Fractions Calculator
Convert two fractions to decimals and subtract to see instantly which is larger, by how much, or whether the two are equal.
In service MI-01-104Compatible Numbers Calculator
Round two numbers to nearby values that are easy to add in your head, then sum those rounded values for a quick mental-math estimate.
In service MI-01-105Complementary Angles Calculator
Find the angle that pairs with yours to complete a right angle: complement = 90° − θ, solved instantly with the subtraction shown in full.
In service MI-01-106Completing the Square Calculator
Convert ax² + bx + c into vertex form a(x + p)² + q — enter the three coefficients and this sheet returns p and q with every step shown.
In service MI-01-107Completing the Square Practice Math Problems and Examples
Transform any quadratic ax²+bx+c into vertex form a(x−p)²+q, showing exactly where each piece of the new form comes from.
In service MI-01-108Complex Conjugate Calculator
Enter a complex number's real and imaginary parts and get its conjugate instantly, with the sign-flip rule and why it matters shown in full.
In service MI-01-109Complex Number Calculator
Convert a complex number a+bi into its modulus (distance from the origin) and argument (angle from the positive real axis).
In service MI-01-110Complex Number to Polar Form Calculator
Convert a complex number a + bi into modulus and argument — r = √(a² + b²), θ = atan2(b, a) — the pair phasor diagrams and AC circuit work run on.
In service MI-01-111Complex Number to Rectangular Form Calculator
Convert a complex number from trigonometric (polar) form to rectangular form: real = r·cos(θ), imaginary = r·sin(θ).
In service MI-01-112Complex Number to Trigonometric Form Calculator
Convert a complex number from rectangular form to trigonometric (polar) form: modulus r = √(real² + imaginary²), argument θ = atan2(imaginary, real).
In service MI-01-113Complex Root Calculator
Solve a quadratic with a negative discriminant for its real part and imaginary magnitude — the conjugate pair a real-coefficient equation forces into existence.
In service MI-01-114Condense Logarithms Calculator
Combine log(x) + log(y) − log(z) into the single logarithm log(xy ⁄ z) — enter three arguments and read back the condensed value.
In service MI-01-115Condition Number Calculator
Measure how sensitive a 2×2 matrix is to small errors — κ = ‖A‖∞ × ‖A⁻¹‖∞, using the infinity-norm largest-row-sum rule on both the matrix and its inverse.
In service MI-01-116Cone Volume Calculator
Turn a cone's base radius and height into volume with V = ⅓πr²h, plus the geometric reason the one-third factor is exact.
In service MI-01-117Conic Sections Calculator
Enter the A, B, and C coefficients of a general conic and B² − 4AC reports whether the curve is an ellipse, a parabola, or a hyperbola.
In service MI-01-118Consecutive Integers Calculator
Enter the sum and the count of a run of consecutive integers to recover the first and last integers, with the algebra shown at every step.
In service MI-01-119Convolution Calculator
Convolve two 2-term sequences directly: the sliding, multiply-and-sum operation behind signal processing and polynomial multiplication.
In service MI-01-120Coordinate Distance Calculator
Find how far a single point sits from the origin — the length (magnitude) of its position vector, via the Pythagorean theorem.
In service MI-01-121Coordinate Grid Calculator
Enter a point's x and y values and see instantly which of the four quadrants — I, II, III, or IV — that pair falls into.
In service MI-01-122Corner Point Calculator
Solve where y = m₁x + b₁ and y = m₂x + b₂ cross — the vertex a feasible region's boundary lines meet, with every step of the algebra shown.
In service MI-01-123Cos 2 Theta Calculator
Enter an angle θ and get cos(2θ) instantly, computed as cos²θ − sin²θ — the double-angle identity, not a doubled cosine value.
In service MI-01-124Cos Inverse Calculator
Find a course's bearing deviation from an adjacent-to-direct-line distance ratio using inverse cosine, a navigation framing of arccosine.
In service MI-01-125Cos-1 Calculator
What the cos⁻¹ button on a scientific calculator actually returns, why it is not 1 ÷ cos(x), and how it matches acos(x) and arccos(x) exactly.
In service MI-01-126Cosecant Calculator
Take the reciprocal of sine to get cosecant, see exactly why it breaks down at 0° and 180°, and where its value can never land between −1 and 1.
In service MI-01-127Cosh Calculator
Evaluate cosh(x) = (eˣ + e⁻ˣ)⁄2 instantly — the unbounded hyperbolic cosine behind catenary curves, relativity's rapidity, and the hyperbola x² − y² = 1.
In service MI-01-128Cosine Calculator
Enter an angle in degrees, radians, or turns and get its exact cosine, worked from the unit-circle definition rather than a black-box lookup.
In service MI-01-129Cosine Similarity Calculator
Score how closely two vectors point the same way with sim = (v1·v2) ⁄ (|v1||v2|), a ratio bounded between −1 and 1.
In service MI-01-130Cosine Triangle Calculator
Enter two sides and the angle trapped between them and this sheet returns the third side with the Law of Cosines, working shown at every step.
In service MI-01-131Cotangent Calculator
Find cot(θ) for any angle in degrees, radians, or turns — the cosine-over-sine ratio, with its blow-ups and zero-crossings explained.
In service MI-01-132Coterminal Angle Calculator
Reduce any angle, however large, negative, or wrapped around, to the single equivalent direction between 0° and 360° with one mod-360 operation.
In service MI-01-133Cramer's Rule Calculator
Solve two linear equations in x and y with Cramer's rule — enter six coefficients and watch each determinant that produces x and y.
In service MI-01-134Cross Multiplication Calculator
Solve any proportion a⁄b = c⁄d for the missing fourth term by cross-multiplying — enter three numbers, get d = bc ⁄ a with the working shown.
In service MI-01-135Cross Product Calculator
Find the vector perpendicular to two 3D arrows — component formula, right-hand-rule direction, and every multiplication shown.
In service MI-01-136Cross-Sectional Area Calculator
Find the area of a ring-shaped cross-section — a pipe wall, washer, or annulus — from its outer and inner radii, A = π(R² − r²).
In service MI-01-137Cube Calc: find V, a, d
Enter one side length and get a cube's volume, surface area, and space diagonal together — three formulas, one shared input, no repeat typing.
In service MI-01-138Cube Calculator
Find a cube's volume and total surface area from a single side length: V = s³ and A = 6s² — every measure determined by that one number.
In service MI-01-139Cube Root Calculator
Find the real cube root of any number, positive, negative, or zero, with the sign carried through exactly instead of erroring out.
In service MI-01-140Cubic Equation Calculator
Test a guessed root of ax³+bx²+cx+d by synthetic division, then solve the leftover quadratic for the remaining two solutions at once.
In service MI-01-141Cubic Feet of a Cylinder Calculator
Find a cylindrical tank, post hole, or pipe's capacity in cubic feet from its radius and height in feet, using V = πr²h.
In service MI-01-142Cubic Inches of a Cylinder Calculator
Find a cylinder's volume in cubic inches from its radius and height, both measured in inches: V = πr²h.
In service MI-01-143Cubic Yards In A Cylinder Calculator
Find a cylinder's volume in cubic yards from its radius and height, both measured in yards: V = πr²h — the unit bulk fill and concrete are ordered in.
In service MI-01-144Cuboid Calculator
Enter a box's length, width, and height to get volume, surface area, and the interior space diagonal together, from a single set of measurements.
In service MI-01-145Cuboid Surface Area Calculator
Total surface area of a rectangular box from its length, width, and height — SA = 2(lw + wh + hl), with each face pair broken out separately.
In service MI-01-146Cuboid Volume Calculator
Multiply length, width, and height to get a cuboid's volume — the capacity inside a box, bin, tank, or room, with every step of the working shown.
In service MI-01-147Cycloid Calculator
Roll a circle along a flat line and this sheet turns the rolling circle's radius into the arch's arc length and the area beneath it.
In service MI-01-148Cyclomatic Complexity Calculator
Turn a control-flow graph's edges, nodes, and connected components into McCabe's cyclomatic complexity score, M = E − N + 2P.
In service MI-01-149Cylinder Volume Calculator
Enter a cylinder's radius and height to get its exact volume, V = πr²h — the base circle's area carried straight up through the height.
In service MI-01-150Cylinder Volume Gallons Calculator
Enter a drum or tank's radius and height in inches to get its exact capacity in US gallons — V = πr²h, divided by 231 cubic inches per gallon.
In service MI-01-151Cylinder Volume Liters Calculator
Turn a cylinder's radius and height straight into liters — the amount a can, bottle, or tank actually holds, worked step by step from V = πr²h.
In service MI-01-152Cylindrical Coordinates Calculator
Convert cylindrical coordinates (r, θ, z) into Cartesian x and y with x = r·cos θ and y = r·sin θ, while height z carries straight across unchanged.
In service MI-01-153Decagon Area Calculator
Find a regular decagon's area from one side length using A = 10s² ⁄ (4·tan 18°) — the ten-triangle formula, shown in full precision.
In service MI-01-154Decagon Calculator
Find a regular decagon's perimeter and area from a single side length: P = 10s, A = 10s² ⁄ (4·tan(π⁄10)).
In service MI-01-155Decimal Calculator
Pull out one digit from a decimal number — tenths, hundredths, or thousandths — using a multiply, floor, and mod-10 sequence shown at every step.
In service MI-01-156Decimal Degrees to Degrees Minutes Seconds Converter
Split a decimal-degree angle into whole degrees, arcminutes, and arcseconds — the D° M′ S″ notation still printed on nautical charts and survey plats.
In service MI-01-157Decimal to Fraction Calculator
Turn any decimal into a fraction in lowest terms by scaling it to a whole number, then dividing by the greatest common divisor.
In service MI-01-158Decimal to Percent Converter
Convert any decimal into its percent form by multiplying by 100, with the decimal-point shift shown and the reverse division back explained.
In service MI-01-159Degrees Minutes Seconds to Decimal Degrees Converter
Combine whole degrees, arcminutes, and arcseconds into one decimal-degree figure — the exact reverse of splitting a decimal angle into D° M′ S″.
In service MI-01-160Descartes' Rule of Signs Calculator
Count the sign changes across a cubic's four coefficients to find the largest possible number of positive real roots, before any factoring.
In service MI-01-161Diagonal of a Rectangle Calculator
Find a rectangle's diagonal from its length and width: d = √(l² + w²), the same identity that squares carpenters' frames and sizes TV screens.
In service MI-01-162Diagonal of a Square Calculator
Enter a square's side length and get its exact corner-to-corner diagonal — d = s√2, with the Pythagorean derivation shown in full.
In service MI-01-163Diagonalize Matrix Calculator
Find a 2×2 matrix's two eigenvalues from its trace and determinant using the same quadratic formula that solves any quadratic equation.
In service MI-01-164Diameter of a Circle Calculator
Enter a circle's radius and get its exact diameter, d = 2r — the one circle conversion that never touches π.
In service MI-01-165Diameter of a Cone Calculator
Work backwards from a cone's volume and height to its base diameter, rearranging V = ⅓πr²h to solve for the radius first.
In service MI-01-166Diameter of a Cylinder Calculator
Enter a cylinder's radius and get the exact diameter of its circular cross-section, d = 2r, independent of its height.
In service MI-01-167Diameter to Radius Calculator
Enter a circle's diameter and get its exact radius, r = d ⁄ 2 — the halving step that area, sector, and shaft formulas actually call for.
In service MI-01-168Diamond Problem Calculator
Enter a sum and a product and get the two numbers that satisfy both — the same step used to factor a quadratic trinomial.
In service MI-01-169Digit Sum Calculator
Add up the digits of any whole number from 0 to 9,999 by pulling each one out with floor-and-remainder arithmetic, then summing the four results.
In service MI-01-170Digital Root Calculator
Reduce any whole number to the single digit that repeated digit-summing eventually reaches, using one mod-9 shortcut instead of looping by hand.
In service MI-01-171Dimensions of a Rectangle Calculator
Recover a rectangle's length and width from just its area and perimeter — the two side lengths hiding inside a quadratic.
In service MI-01-172Direct Variation Calculator
Enter a known x, y pair and a new x-value to find the constant of variation k and the matching y — y = kx, worked step by step.
In service MI-01-173Direction of the Vector Calculator
Find which way a 2D vector points, measured from the positive x-axis, using atan2(y, x) — correctly placed in the right quadrant automatically.
In service MI-01-174Discriminant Calculator
Compute the discriminant b² − 4ac of any quadratic and see instantly whether it has two, one, or no real roots.
In service MI-01-175Distance Between Two Points Calculator
Find the straight-line distance between two coordinate points using the Distance Formula: d = √((x₂−x₁)² + (y₂−y₁)²), the Pythagorean theorem in disguise.
In service MI-01-176Distance Formula Calculator
Find the straight-line distance between two points in space, each given as (x, y, z) — the Pythagorean theorem extended one dimension further.
In service MI-01-177Distance from Point to Plane Calculator
Measure the shortest, perpendicular distance from any point to a plane written as Ax+By+Cz+D=0, with the normal-vector math shown.
In service MI-01-178Distributive Property Calculator
Expand a(b + c) into ab + ac, or check the two sides agree — the rule that connects multiplication to addition, with the gap always at zero.
In service MI-01-179Divide Complex Numbers Calculator
Divide (a + bi) by (c + di) using the conjugate trick that clears the imaginary denominator, with the real and imaginary results shown exactly.
In service MI-01-180Dividing Exponents Calculator
Divide two powers of the same base by subtracting exponents — aᵐ ⁄ aⁿ = aᵐ⁻ⁿ, worked out with the sign and every step shown.
In service MI-01-181Dividing Fractions Calculator
Divide one fraction by another with invert-and-multiply, (a/b) ÷ (c/d) = ad/bc, showing the reciprocal step plainly.
In service MI-01-182Dividing Radicals Calculator
Combine two square roots into one with √a ⁄ √b = √(a ⁄ b), then read the simplified quotient — no separate decimal lookups needed.
In service MI-01-183Divisibility Test Calculator
Test whether one whole number divides another by computing the remainder of a mod n directly — a remainder of zero means n divides a exactly.
In service MI-01-184Division Calculator
Split one number by another and get both pieces of the answer — the whole-number quotient and the leftover remainder — checked against a = b×q + r.
In service MI-01-185Divisor Calculator
Test whether one whole number divides evenly into another — a remainder of zero means yes, the formal definition behind 'goes into evenly.'
In service MI-01-186Dodecagon Area Calculator
Enter a regular dodecagon's side length and get its exact area, worked from twelve identical triangles meeting at the shape's center.
In service MI-01-187Dodecagon Calculator
Find the area of a regular twelve-sided dodecagon from one side length, using the exact closed-form value hidden inside its formula.
In service MI-01-188Does completing the square always work for quadratic equations?
Test whether a quadratic has real roots by checking its discriminant — the number that decides what completing the square actually reveals.
In service MI-01-189Dot Product Calculator
Multiply matching components of two vectors and add the results — one signed number that shows whether they align, oppose, or sit at a right angle.
In service MI-01-190Double Angle Calculator
Turns any angle θ into sin(2θ) using the double-angle identity 2 sinθ cosθ, skipping the step of forming 2θ and looking up its sine separately.
In service MI-01-191Double Angle Formula Calculator
Enter an angle once and read sin(2θ), cos(2θ), and tan(2θ) together — all three double-angle identities evaluated from a single θ.
In service MI-01-192Double Angle Identities Calculator
See sin(2θ) computed two independent ways, a double-angle shortcut and a general angle-addition formula, then confirm both land on one identical number.
In service MI-01-193Doubling Time Calculator
Give it a steady growth rate and get the exact number of periods a quantity takes to double, using t = ln(2) ⁄ r.
In service MI-01-194e Calculator | eˣ | e Raised to Power of x
Raise Euler's number e to any exponent x and read off eˣ instantly — the natural exponential function, its own derivative, worked to full precision.
In service MI-01-195Egyptian Fractions Calculator
Break a proper fraction into a sum of distinct unit fractions, ancient Egypt's own notation, showing the first two greedy terms.
In service MI-01-196Eigenvalue and Eigenvector Calculator
Solve a 2×2 matrix's characteristic equation for its two eigenvalues, then walk through finding the actual eigenvector each one belongs to.
In service MI-01-197Elimination Method Calculator
Solve two linear equations for x and y by scaling one until a variable's coefficients match, then subtracting — the classic hand-elimination technique.
In service MI-01-198Ellipse Area Calculator
Find the surface an ellipse encloses from its two semi-axis lengths — A = πab, an exact formula with no approximation involved.
In service MI-01-199Ellipse Calculator
Find an ellipse's area exactly (A = πab) and its perimeter via Ramanujan's close approximation, from the two semi-axis lengths.
In service MI-01-200Ellipse Circumference Calculator
Enter an ellipse's two semi-axes and get its circumference from Ramanujan's 1914 approximation, with every step of the arithmetic shown.
In service MI-01-201Ellipse Perimeter Calculator
Enter an ellipse's semi-major and semi-minor axes to get its perimeter from Ramanujan's 1914 approximation, with every step shown.
In service MI-01-202Ellipse Standard Form Calculator
Test any point against an ellipse's standard-form equation and see at once whether it falls inside, exactly on, or outside the curve.
In service MI-01-203Ellipsoid Volume Calculator
Turn three semi-axis lengths into an ellipsoid's volume with V = (4⁄3)πabc, the sphere formula generalized to three independent radii.
In service MI-01-204Endpoint Calculator
Give a segment's midpoint and one endpoint, and this sheet solves algebraically for the endpoint you don't know — no graph paper required.
In service MI-01-205Engineering Notation Calculator
Convert a number into mantissa times ten to a power, with the exponent locked to a multiple of three so it lines up with metric prefixes.
In service MI-01-206Equation of a Circle Calculator
Test whether a point lies on, inside, or outside a circle using its standard equation: (x−h)² + (y−k)² compared against r².
In service MI-01-207Equation of a Circle with Diameter Endpoints Calculator
Turn two diameter endpoints into a circle's center, radius, and full equation — the midpoint and half-distance formulas, worked exactly.
In service MI-01-208Equation of a Plane Calculator
Turn any three non-collinear points in space into the plane's equation Ax + By + Cz = D, using the cross product to find the normal vector directly.
In service MI-01-209Equation of a Sphere Calculator
Enter a sphere's center and radius plus any test point; (x−h)² + (y−k)² + (z−l)² − r² returns 0 on the surface, negative inside, positive outside.
In service MI-01-210Equilateral Triangle Area Calculator
Enter one side length and get the enclosed area of an equilateral triangle, with the (√3⁄4)s² shortcut derived and checked against Heron's formula.
In service MI-01-211Equilateral Triangle Calculator
Find every basic measure of an equilateral triangle from one side length: area, perimeter, and height, all determined by that single number.
In service MI-01-212Equivalent Fractions Calculator
Scale a fraction's numerator and denominator by the same factor k to generate an equivalent fraction, with the multiplication shown plainly.
In service MI-01-213Equivalent Ratio Calculator
Scale a known ratio to a new first term and get the exact matching second term, with the cross-multiplication shown alongside it.
In service MI-01-214Error Function Calculator
Estimate erf, tied directly to the normal distribution's cumulative probability, from four Maclaurin-series terms, with the small truncation gap shown honestly.
In service MI-01-215Euclidean Algorithm Calculator
Trace the Euclidean algorithm's repeated division step by step and watch the greatest common divisor fall out of the last nonzero remainder.
In service MI-01-216Euclidean Distance Calculator
Find the straight-line distance between two points on a plane using the Pythagorean theorem, the simplest introductory case.
In service MI-01-217Euler’s Formula for Polyhedron Calculator
Find a solid's face count from its vertices and edges with V − E + F = 2, the identity holding for any polyhedron shaped like a sphere.
In service MI-01-218Exact Value of Trig Functions Calculator
Look up sine and cosine's exact, non-decimal values at the five standard special angles: 0°, 30°, 45°, 60°, and 90°.
In service MI-01-219Expanded Form Calculator
Break any whole number from 0 to 9,999 into its thousands, hundreds, tens, and ones place values — see exactly what each digit is worth.
In service MI-01-220Expanding Logarithms Calculator
Split log(a × b) into ln(a) + ln(b) and watch both forms return the same figure — the product rule for logarithms, shown side by side.
In service MI-01-221Exponent Calculator
Multiply a base by itself as many times as the exponent says — whole, zero, fractional, or negative — with the exact result shown.
In service MI-01-222Exponential Form Calculator
Convert a discrete growth base b (from y = a·bˣ) to the continuous rate k needed for y = a·e^(kx): k = ln(b).
In service MI-01-223Exponential Function Calculator
Evaluate f(x) = a·bˣ for any coefficient, base, and exponent, and watch the multiplication unfold one step at a time.
In service MI-01-224Exponential Growth Calculator
Compound a starting value by a fixed percentage rate over repeated periods with P(t) = P₀(1+r)ᵗ, shown step by step.
In service MI-01-225Exponential Notation Calculator
Normalize any nonzero number into mantissa × 10ⁿ form, with the exponent and mantissa worked out step by step from log₁₀|x|.
In service MI-01-226Exterior Angles of a Triangle Calculator
Add the two remote interior angles to get a triangle's exterior angle at the third vertex — Euclid's Proposition 32, computed exactly.
In service MI-01-227Factor Calculator
Factor a² − b² instantly into (a − b)(a + b), the classic difference-of-squares pattern, with both resulting factors shown at once.
In service MI-01-228Factorial Calculator
Multiplies n × (n−1) × … × 1 to find n factorial, the count of every possible ordering of n distinct objects, with each step shown plainly.
In service MI-01-229Factoring Trinomials Calculator
Factor any trinomial ax²+bx+c by finding its two roots with the quadratic formula, revealing its binomial factors.
In service MI-01-230Fermat's Little Theorem Calculator
Enter a base and a prime and this sheet raises the base to the power p − 1, reduces it modulo p, and lands on 1 — Fermat's little theorem, worked instantly.
In service MI-01-231Fibonacci Calculator
Enter a term index n and get the nth Fibonacci number instantly, computed directly from the golden ratio — no adding up the sequence by hand.
In service MI-01-232Fifth Root Calculator
Find the real fifth root of any number, positive, negative, or zero, with the sign preserved and no complex result or error along the way.
In service MI-01-233Floor Division Calculator
Divide two numbers and keep only the whole number of times one fits into the other — ⌊a ⁄ b⌋ — remainder discarded, negatives handled correctly.
In service MI-01-234Floor Function Calculator
Enter any real number and get its floor, the greatest integer at or below it, with the rounding direction shown clearly for negatives too.
In service MI-01-235Foci of an Ellipse Calculator
Find how far an ellipse's two foci sit from its center, given the semi-major and semi-minor axes, using c = √(a² − b²).
In service MI-01-236FOIL Calculator
Expand (ax + b)(cx + d) into its four term-by-term products — First, Outer, Inner, and Last — and see exactly which multiplication produced each one.
In service MI-01-237Fourth Root Calculator
Find the fourth root of any non-negative number — the value that, raised to the fourth power, reproduces x exactly, with the reasoning shown.
In service MI-01-238Fraction Calculator
Perform any basic operation on two fractions and get the result already simplified to lowest terms, with an operation selector.
In service MI-01-239Fraction Exponent Calculator
Evaluate x raised to a fractional exponent m⁄n by taking the nth root of the base and raising that root to the mth power, with every step shown.
In service MI-01-240Fraction To Decimal Converter
Convert any fraction a⁄b into its decimal value through ordinary long division, and see exactly why some fractions terminate while others repeat forever.
In service MI-01-241Fraction to Percent Calculator
Convert any fraction a⁄b into an exact percentage. Multiply by 100 and watch the arithmetic that turns a ratio into a percent.
In service MI-01-242Fundamental Counting Principle Calculator
Multiply the choices available at each step of a sequence — n₁ × n₂ × n₃ — to find every possible combined outcome without listing them by hand.
In service MI-01-243Galileo's Paradox of Infinity Calculator
Pair a counting number with its perfect square — the exact bijection at the heart of Galileo's centuries-old paradox about infinite sets.
In service MI-01-244Gamma Function Calculator
Evaluates the gamma function at a positive whole number using Γ(n) = (n−1)!, the exact integer case of Euler's continuous extension of the factorial.
In service MI-01-245Gauss-Jordan Elimination Calculator
Fully solve a 2-equation system by reducing both rows until x and y are isolated directly, no back-substitution needed.
In service MI-01-246GCD Calculator
Find the greatest common divisor of two whole numbers — the largest number that divides both evenly, the key to reducing any fraction to lowest terms.
In service MI-01-247GCF and LCM Calculator
Enter two whole numbers once and this sheet returns both the greatest common factor and the least common multiple, linked by one identity.
In service MI-01-248GCF Calculator – Greatest Common Factor
Find the greatest common factor shared by a fraction's numerator and denominator, then reduce that exact fraction to lowest terms in one pass.
In service MI-01-249General Form of the Equation of a Circle Calculator
Enter a circle's center and radius to get D, E, and F — the coefficients in x² + y² + Dx + Ey + F = 0, expanded and checked term by term.
In service MI-01-250General to Standard Form of a Circle Calculator
Convert a circle's general-form equation, x² + y² + Dx + Ey + F = 0, to standard form by completing the square: recover the center and radius directly.
In service MI-01-251Generic Rectangle Calculator
Multiply (ax+b) by (cx+d) using the area-model rectangle: four sub-areas combine into the x² coefficient, the x coefficient, and the constant.
In service MI-01-252Geometric Mean Calculator
Multiply two non-negative numbers, take the square root, and get the average built for ratios, rates, and compounding growth — not a simple sum-and-halve mean.
In service MI-01-253Geometric Sequence Calculator
Enter a first term and common ratio to find any term in a geometric sequence, with the exponent and every multiplication shown plainly.
In service MI-01-254Golden Ratio Calculator
Divide any length into a larger and smaller part so the whole is to the larger part as the larger is to the smaller — the golden ratio, φ = (1 + √5) ⁄ 2.
In service MI-01-255Golden Rectangle Calculator
Enter one side of a golden rectangle and get the matching long side and area, worked from long = short × φ with every step of the arithmetic shown.
In service MI-01-256Gradient Calculator
Enter two coordinate points and get the gradient — the constant rise-over-run ratio between them — with every step of the division shown.
In service MI-01-257Gram-Schmidt Calculator
Subtract v₂'s projection onto v₁ and what's left, u₂, is perpendicular to v₁ — the Gram-Schmidt step worked through component by component.
In service MI-01-258Graphing Inequalities on a Number Line Calculator
Plug a candidate number into ax + b < c and see whether it lands in the shaded region, sits right on the boundary, or falls outside it entirely.
In service MI-01-259Graphing Quadratic Inequalities Calculator
Check whether a point sits above, below, or exactly on a parabola y=ax²+bx+c — the one subtraction behind shading a quadratic inequality's correct region.
In service MI-01-260Great Circle Calculator
Find the shortest distance between two points on Earth along the planet's curved surface, using the haversine formula and mean Earth radius.
In service MI-01-261Greater Than Or Less Than Calculator
Compare two numbers directly: a − b — positive means the first is greater, negative means it's less, and zero means they're equal.
In service MI-01-262Greatest Common Denominator Calculator
Find the largest group size that splits two batches — tile counts, item counts, anything countable — into equal groups with nothing left over.
In service MI-01-263Greatest Common Divisor Calculator
Find the greatest common divisor shared by three numbers at once, extending the two-number case one step further.
In service MI-01-264Greatest Common Factor Calculator
Break two whole numbers into prime factors, spot the primes they share, and multiply those shared primes to reach the greatest common factor.
In service MI-01-265Hadamard Product Calculator
Multiply two matrices entry by entry, position against matching position, instead of row against column — every product cell shown.
In service MI-01-266Half Angle Calculator
Feed in a full angle's cosine and get sin(θ⁄2) and cos(θ⁄2) at once — the half-angle identities solved from cos(2x), worked out exactly.
In service MI-01-267Harmonic Mean Calculator
Average two rates the honest way: the harmonic mean of a and b, computed as 2ab ⁄ (a+b), shown alongside the arithmetic and geometric means.
In service MI-01-268Harmonic Number Calculator
Add three unit fractions, 1/a + 1/b + 1/c, and see sequential denominators 1, 2, 3 produce the exact third harmonic number, 11/6.
In service MI-01-269Height of a Cone Calculator
Rearrange a cone's volume formula to solve for its height when the capacity and base radius are already known: h = 3V ⁄ (πr²).
In service MI-01-270Height of a Cylinder Calculator
Recover a cylinder's height from its volume and radius by inverting V = πr²h — pure division, with no cube root or one-third factor involved.
In service MI-01-271Height of a Square Pyramid Calculator
Solve for a square pyramid's height from its volume and base side length: h = 3V ⁄ s², the volume formula rearranged and verified.
In service MI-01-272Heptagon Area Calculator
Compute a regular heptagon's area from one side length using A = 7s² ⁄ (4·tan(π⁄7)), with every step of the seven-triangle formula shown.
In service MI-01-273Heptagon Calculator
Find a regular heptagon's perimeter and area from a single side length: P = 7s, A = 7s² ⁄ (4·tan(π⁄7)).
In service MI-01-274Heron's Formula Calculator
Find any triangle's area from its three side lengths alone using Heron's Formula: A = √(s(s−a)(s−b)(s−c)), no angle or height needed.
In service MI-01-275Hexagon Calculator
Turn one side length into a regular hexagon's area and perimeter — A = (3√3 ⁄ 2)s², split cleanly into six equilateral triangles.
In service MI-01-276Hexagonal Pyramid Calculator
Find a hexagonal pyramid's total surface area from its base edge and slant height: base area plus six identical triangular faces.
In service MI-01-277Hexagonal Pyramid Surface Area Calculator
Find a hexagonal pyramid's base area, its six triangular faces, and total surface area from side length and slant height, with every step shown.
In service MI-01-278Hilbert's Hotel Paradox Calculator
Find the new room number for a guest after new arrivals shift every guest over — the classic thought experiment about infinity, made concrete.
In service MI-01-279Hollow Cylinder Volume Calculator
Find the material volume of a pipe or tube — the outer cylinder minus the inner cavity, V = π(R² − r²)h, computed exactly.
In service MI-01-280Hyperbolic Functions Calculator
Enter one x and get sinh(x), cosh(x), and tanh(x) together — three functions built from the same eˣ and e⁻ˣ pair, reported at once.
In service MI-01-281Hypotenuse Calculator
Enter the two legs of a right triangle and get the exact hypotenuse, c = √(a² + b²), with the squaring, adding, and rooting steps shown.
In service MI-01-282i Calculator
Raise the imaginary unit i to a whole-number power and read off the real and imaginary parts of the four-step cycle it produces.
In service MI-01-283Imaginary Number Calculator
Take the square root of any negative number and read off its imaginary coefficient, with the reasoning behind i² = −1 explained plainly.
In service MI-01-284Improper Fraction to Mixed Number Calculator
Convert an improper fraction into a whole number and a leftover fraction over the same denominator, using one floor and one subtraction.
In service MI-01-285Inequality to Interval Notation Calculator
Solve ax + b < c step by step, isolating x the same way you would solve the matching equation, then read the result off as interval notation.
In service MI-01-286Inscribed Angle Calculator
Halve a central angle into the inscribed reading for the same arc, or double an inscribed value back to the center — the Inscribed Angle Theorem, exact.
In service MI-01-287Integer Calculator
Subtract a number's floor from itself to see what is left over — zero means a genuine integer, and anything above zero means a fractional part remains.
In service MI-01-288Integration by Completing the Square and Substitution
Complete the square under 1⁄(x²+bx+c) to reach the standard arctangent antiderivative, with the shift, the constant, and the leading coefficient shown.
In service MI-01-289Interior and Exterior Triangle Angles Calculator
Find a triangle's exterior angle from its interior angle: exterior = 180° − interior — the two always sit together on a straight line.
In service MI-01-290Intersection of Two Lines Calculator
Find where two lines in slope-intercept form cross: x = (b₂−b₁) ⁄ (m₁−m₂), then y from either line's own equation.
In service MI-01-291Interval Notation Calculator
Learn how closed, open, and half-open intervals are written, then see how any interval's width and midpoint are computed from its two endpoints.
In service MI-01-292Inverse Cosine Calculator
Find the angle whose cosine equals a given value: acos(x), defined only for x between −1 and 1, always returning a result between 0° and 180°.
In service MI-01-293Inverse Matrix Calculator
Divide a 2×2 matrix's adjoint by its determinant to find the inverse — enter a, b, c, and d and get all four inverse entries, or a singular-matrix flag.
In service MI-01-294Inverse Modulo Calculator
Find the modular inverse of a number under a prime modulus using Fermat's Little Theorem — one exponentiation and one remainder, no repeated-squaring loop.
In service MI-01-295Inverse Sine Calculator
Find the angle whose sine equals a given value: asin(x), defined only for x between −1 and 1, the exact range plain sine itself produces.
In service MI-01-296Inverse Tangent Calculator
Find the angle whose tangent equals a given value: atan(x) — unlike inverse sine or cosine, defined for every real number, since tangent itself is unbounded.
In service MI-01-297Inverse Trigonometric Functions Calculator
See asin, acos, and atan evaluated together at the same input — a side-by-side comparison of how the three inverse trig functions actually differ.
In service MI-01-298Inverse Variation Calculator
Solve y = k ⁄ x for a constant of variation and an input, or recover the constant from a known pair — the relationship where x·y never changes.
In service MI-01-299Involute Function Calculator
Unwind a taut string from a circle and track where its end lands — the x and y coordinates of the involute, the curve that shapes every gear tooth.
In service MI-01-300Irregular Polygon Area Calculator
Find the area of an irregular four-sided figure directly from its four corner coordinates, using the Shoelace Formula.
In service MI-01-301Irregular Trapezoid Area Calculator
Solve a trapezoid's height and area when both legs differ in length, with no assumption that either one matches the other before you begin.
In service MI-01-302Is it a Right Triangle Calculator
Enter three side lengths and get a single yes-or-no answer for whether they form a right triangle — no protractor, no guessing which side is longest.
In service MI-01-303Is Modulo Multiplication and Addition Associative, Distributive, and Commutative?
Verify that (a + b) mod m always equals ((a mod m) + (b mod m)) mod m — the identity that lets remainder arithmetic be reduced piece by piece.
In service MI-01-304Isosceles Right Triangle Calculator
Enter one leg of a 45-45-90 triangle and get the hypotenuse and the enclosed area together, both worked exactly from that single length.
In service MI-01-305Isosceles Right Triangle Hypotenuse Calculator
Find the hypotenuse of an isosceles right triangle from just one equal leg: hypotenuse = leg × √2 — the 45-45-90 triangle's fixed ratio.
In service MI-01-306Isosceles Trapezoid Area Calculator
Find an isosceles trapezoid's area from its two parallel bases and one equal leg length — the height derived automatically via the Pythagorean theorem.
In service MI-01-307Isosceles Trapezoid Calculator
Add two parallel bases to twice one shared leg length and get an isosceles trapezoid's full perimeter in a single quick step, no height involved.
In service MI-01-308Isosceles Triangle Angles Calculator
Find the two equal base angles and the apex angle of an isosceles triangle from its base and leg lengths, with the inverse-cosine step shown.
In service MI-01-309Isosceles Triangle Area Calculator
Enter the base and the equal leg length of an isosceles triangle and get its exact area, with the height shown as the intermediate step.
In service MI-01-310Isosceles Triangle Calculator
Solve an isosceles triangle completely from its base and one equal leg: height, area, and perimeter all returned together.
In service MI-01-311Isosceles Triangle Find A Calculator
Find an isosceles triangle's vertex angle — the angle between its two equal legs — from the leg length and the base: 2·asin((base⁄2) ⁄ a).
In service MI-01-312Isosceles Triangle Height Calculator
Drop the altitude from an isosceles triangle's apex to its base and read the exact height from the base and one equal leg, worked step by step.
In service MI-01-313Isosceles Triangle Side Calculator
Find the equal leg length of an isosceles triangle from its base and area — the reverse direction of the standard height-and-area calculation.
In service MI-01-314Kite Area Calculator
Enter a kite's two diagonals and get the enclosed area at once — A = ½·d₁·d₂, exact with no angle to measure, since the crossing is always 90°.
In service MI-01-315Lagrange Error Bound Calculator
Cap how far a Taylor polynomial can stray from the true value, using only a ceiling M on the next derivative's size — no exact remainder needed.
In service MI-01-316Lateral Area of a Cone Calculator
Find the curved side area of a cone from its radius and slant height alone, using A = πrl — the base deliberately left out.
In service MI-01-317Lateral Area Trapezoidal Prism Calculator
Enter a trapezoid base's four edges and the prism's length to get lateral surface area from A = perimeter × length — no base area, no height needed.
In service MI-01-318Lateral Surface of a Cylinder Calculator
Find a cylinder's curved side area from radius and height using A = 2πrh — the wraparound surface only, without the top and bottom caps.
In service MI-01-319Latus Rectum Calculator
Enter a from y² = 4ax and get the latus rectum instantly — the focal chord's exact length, derived from the two points where it meets the curve.
In service MI-01-320Law of Cosines Calculator
Find a triangle's angle from all three known side lengths using the Law of Cosines: C = acos((a² + b² − c²) ⁄ 2ab), the Pythagorean theorem generalized.
In service MI-01-321Law of Sines Calculator
Find a triangle's missing side from one known side-angle pair and one more angle: b = a·sinB ⁄ sinA — the ratio of side to opposite sine stays constant.
In service MI-01-322LCD Calculator - Least Common Denominator
Enter two fraction denominators and get the least common denominator — the shared base you need before adding or subtracting the fractions.
In service MI-01-323LCM Calculator — Least Common Multiple
Find the least common multiple of two whole numbers — the smallest number both divide into evenly, the key to adding fractions with different denominators.
In service MI-01-324Least Common Denominator Calculator
Add three fractions at once — enter all three bottom numbers and get the smallest shared denominator every one of them converts into cleanly.
In service MI-01-325Least Common Factor Calculator
Enter two gears' tooth counts and find how many meshes pass before their marked teeth line back up — the least common multiple in mechanical form.
In service MI-01-326Least Common Multiple Calculator
Find the smallest number that three separate numbers all divide into evenly, extending the two-number LCM one step further.
In service MI-01-327Least Squares Regression Line Calculator
Enter three x,y points and get the best-fit line's slope and intercept — the pair that minimizes the total squared vertical miss, worked step by step.
In service MI-01-328Length and Width of a Rectangle Given Area Calculator
Recover BOTH a rectangle's dimensions from just its area and perimeter — solved as the two roots of a quadratic equation.
In service MI-01-329Length and Width of a Rectangle given Perimeter Calculator
Recover a rectangle's missing side from its perimeter and one known length — width = perimeter ÷ 2 minus length, a single subtraction.
In service MI-01-330Length of a Line Segment Calculator
Finds a segment's midpoint — its exact center — from two endpoints. For the segment's actual length, see this site's distance calculators instead.
In service MI-01-331Length Of A Rectangle Calculator
Recover a rectangle's missing side by dividing area by width — the algebraic flip of A = l × w, solved and checked in one step.
In service MI-01-332LFSR Calculator
Run one step of a 4-bit linear-feedback shift register, shifting the bits and feeding back a new one via XOR at two tap positions.
In service MI-01-333Line Equation from Two Points Calculator
Enter two points and get the line's slope and y-intercept together — the two numbers that fully describe y = mx + b.
In service MI-01-334Line of Intersection of Two Planes Calculator
Cross two planes' normal vectors to find the direction of the line where they meet, or the zero vector if the planes run parallel.
In service MI-01-335Linear Combination Calculator
Scale two vectors by their own coefficients and add the results — c₁v₁ + c₂v₂ — the operation behind every span and basis.
In service MI-01-336Linear Independence Calculator
Test whether two 2D vectors are linearly independent or dependent from one determinant, v1x·v2y − v1y·v2x, with every step shown.
In service MI-01-337Linear Interpolation Calculator
Estimate an in-between y-value from two known points, x₁,y₁ and x₂,y₂ — the straight-line assumption behind reading a value off a table.
In service MI-01-338Log Base 2 Calculator
Find the exponent that turns 2 into x — enter any positive number and read log₂(x), the base-2 logarithm, computed exactly.
In service MI-01-339Log Calculator (Logarithm)
Compute logₐ(x) for any base a, using the change-of-base identity: logₐ(x) = ln(x) ⁄ ln(a) — one formula, every base.
In service MI-01-340Long Addition Calculator
Add two two-digit numbers column by column and see exactly when and why a carry moves into the next column.
In service MI-01-341Long Multiplication Calculator
Multiply a two-digit number by a single digit and see the ones and tens partial products before they combine into the total.
In service MI-01-342Long Subtraction Calculator
Subtract one two-digit number from another and see exactly when and why a column has to borrow from its neighbor.
In service MI-01-343Lowest Common Denominator Calculator
Convert two fractions onto their lowest common denominator and instantly see which one is larger, the comparison a shared denominator exists to make possible.
In service MI-01-344Lowest Term Calculator
Reduce any fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor, in one step.
In service MI-01-345LU Decomposition Calculator
Split a 2×2 matrix into a lower-triangular L and upper-triangular U such that L×U reconstructs the original exactly.
In service MI-01-346Luhn Algorithm Calculator
Run the Luhn checksum used to catch card and ID typos on an 8-digit number: double every second digit from the right, sum, and test for a multiple of 10.
In service MI-01-347Magic Square Calculator
Enter nine numbers into a 3×3 grid and check whether every row, column, and diagonal adds up to the same total — the classic magic square test.
In service MI-01-348Manhattan Distance Calculator
Add the horizontal and vertical coordinate gaps between two points — no diagonal shortcuts — to get the taxicab distance across a grid.
In service MI-01-349Math Power Calculator
Raise any base to any real exponent and read the result exactly — the single calculation behind this site's narrower power sheets.
In service MI-01-350Matrix Addition and Subtraction Calculator
Add two 2×2 matrices entry by entry, or type negative values to subtract — every summed cell shown beside the position-matched formula.
In service MI-01-351Matrix by Scalar Calculator
Multiply a 2×2 matrix by a single scalar k, stretching every entry by the same amount — four products shown, plus how the determinant reacts.
In service MI-01-352Matrix Calculator
Find a 2×2 matrix's determinant and trace: det = ad−bc, trace = a+d — two fundamental numbers summarizing the whole matrix.
In service MI-01-353Matrix Determinant Calculator
Enter a 2×2 matrix's four entries and get its determinant, ad − bc — the signed area factor behind invertibility, area scaling, and orientation.
In service MI-01-354Matrix Multiplication Calculator
Multiply two 2×2 matrices the way linear algebra actually defines it: each row of the first against each column of the second, dot product shown.
In service MI-01-355Matrix Norm Calculator
Compute the Frobenius norm of a 2×2 matrix — square every entry, sum them, and take the square root, the way you'd measure a vector's length.
In service MI-01-356Matrix Power Calculator
Square a 2×2 matrix through genuine row-times-column multiplication, A·A — not by squaring each entry, which is a different operation.
In service MI-01-357Matrix Rank Calculator
Find a 2×2 matrix's rank — 2 if invertible, 1 if its rows collapse to a single direction, 0 only for the all-zero matrix.
In service MI-01-358Matrix Trace Calculator
Add the two diagonal entries of a 2×2 matrix — a and d — to get the trace; the off-diagonal entries b and c never factor in.
In service MI-01-359Matrix Transpose Calculator
Transpose a 2×2 matrix by flipping every entry across the main diagonal — rows become columns, columns become rows.
In service MI-01-360Mayan Numerals Converter
Convert any whole number from 0 to 399 into the ancient Mayan base-20 system: two vigesimal places, a high count of twenties and a low remainder.
In service MI-01-361Mean Calculator
Enter three values and read their arithmetic mean instantly — sum divided by count, with the formula and the reverse-solve line shown.
In service MI-01-362Midpoint Calculator
Average two points' x- and y-coordinates to land on the exact center of the segment between them — the midpoint formula, step by step.
In service MI-01-363Midsegment of a Trapezoid Calculator
Find a trapezoid's midsegment, the average of its two parallel bases, with the triangle-midsegment link and a fully worked example.
In service MI-01-364Midsegment of a Triangle Calculator
Enter a triangle's third side and get its midsegment instantly — exactly half that side's length, guaranteed by the Midsegment Theorem.
In service MI-01-365Miracle Calculator
Multiply a self-rated Faith score by a self-rated Extraordinary Effort score, Hal Elrod's framework for weighing belief and effort together.
In service MI-01-366Missing Side of a Right Triangle Calculator
Enter the hypotenuse and one leg of a right triangle to get the exact length of the missing leg, with every squaring and rooting step shown.
In service MI-01-367Mixed Number Calculator
Convert an improper fraction to a mixed number: whole part = ⌊numerator ⁄ denominator⌋, remainder becomes the new numerator over the same denominator.
In service MI-01-368Mixed Number to Improper Fraction Calculator
Turn a whole number and a proper fraction into a single improper fraction numerator over the same denominator.
In service MI-01-384Möbius Strip Calculator
Compute the flat surface area of a Möbius strip — simply length times width — and see why the half-twist changes its topology, not its area.
In service MI-01-369Modulo Calculator
Divide a by b and keep only the remainder — the wraparound arithmetic behind clocks, hashing, and even/odd checks.
In service MI-01-370Modulo in the Order of Operations
See how a + (b mod c) and (a + b) mod c can give genuinely different answers — modulo shares precedence with multiplication, ahead of addition.
In service MI-01-371Modulo Operations with Negative Numbers
Finds a mod b when a is negative, using the floored convention where the remainder always takes the sign of b — the source of most 'negative modulo' confusion.
In service MI-01-372Modulo Operator: Practical Uses in Arithmetics
Turn a raw minute count into whole hours plus a remainder — floor division finds the hours, and this sheet's other operation finds what's left over.
In service MI-01-373Moment of Inertia Calculator
Combine two point masses and their distances from a shared axis into one moment of inertia, I = m₁r₁² + m₂r₂² — the sum shown term by term.
In service MI-01-374Multiplication Calculator
Multiply two numbers — whole, negative, or decimal — and get the exact product, with the distributive law and sign rules behind it shown.
In service MI-01-375Multiplicative Inverse Calculator
Find the multiplicative inverse of any nonzero number — enter x and read 1 ⁄ x, with the identity x × (1 ⁄ x) = 1 confirmed alongside it.
In service MI-01-376Multiplicative Inverse Modulo Calculator
Finds a number's multiplicative inverse modulo a prime via Fermat's Little Theorem — the same step RSA-style key generation relies on internally.
In service MI-01-377Multiply Complex Numbers Calculator
Expand (a + bi) times (c + di) into one product — real part ac − bd and imaginary part ad + bc, both computed and shown together.
In service MI-01-378Multiplying Binomials Calculator
Multiply (ax + b)(cx + d) into one simplified trinomial — ac, ad + bc, and bd already combined, one step past FOIL's four separate terms.
In service MI-01-379Multiplying Exponents Calculator
Multiply two powers of the same base by adding their exponents — aᵐ × aⁿ = aᵐ⁺ⁿ, worked out with every step shown.
In service MI-01-380Multiplying Fractions Calculator
Multiply two fractions straight across — numerator times numerator, denominator times denominator — with the unreduced product shown plainly.
In service MI-01-381Multiplying Polynomials Calculator
Multiply a binomial by a trinomial and get all four coefficients of the resulting cubic, with every one of the six cross-terms shown before they combine.
In service MI-01-382Multiplying Radicals Calculator
Multiply two square roots into one with √a × √b = √(ab), then read the exact simplified product instead of two separate rounded decimals.
In service MI-01-383Multiplying Scientific Notation Calculator
Multiply two numbers written as mantissa times a power of ten by multiplying the mantissas and adding the exponents, renormalized automatically.
In service MI-01-385Natural Log Calculator
Compute ln(x), the natural logarithm base e, with the inverse check and the log-10 conversion constant shown.
In service MI-01-386Negative Log Calculator
Flip the sign of a base-10 logarithm to turn a small fraction into a clean positive figure — the arithmetic behind pH and decibel scales.
In service MI-01-387NOR Calculator
Enter A and B to get A NOR B, the negation of OR: 1 only when both inputs are 0, and 0 in the other three cases, with the arithmetic shown.
In service MI-01-388Null Space Calculator
Find a 2×2 matrix's nullity (the dimension of its null space) from its rank: nullity = 2 − rank, the Rank-Nullity Theorem in action.
In service MI-01-389Oblique Triangle Calculator
Enter two sides and the angle between them; the Law of Cosines returns the third side — the general formula that works even with no right angle.
In service MI-01-390Obtuse Triangle Calculator
Test whether a triangle is obtuse, right, or acute from its three side lengths — a²+b²−c², the Law of Cosines' own sign, applied as a classifier.
In service MI-01-391Octagon Calculator
Turn one side length into a regular octagon's area and perimeter — A = 8s² ⁄ (4·tan(π⁄8)), a formula that collapses to a clean, trig-free constant.
In service MI-01-392One's Complement Calculator
Find a 4-bit number's one's complement by flipping every bit, computed directly as 15 minus the number.
In service MI-01-393OR Calculator
Add two 0-or-1 inputs and subtract their overlap to get logical OR — true whenever at least one side holds, shown row by row.
In service MI-01-394Order of Magnitude Calculator
Enter any positive number and read its order of magnitude — the power-of-ten decade it has cleared, found by flooring log₁₀(x).
In service MI-01-395Orthocenter Calculator
Enter a triangle's three vertices and get the orthocenter — where its three altitudes meet, built from the centroid and circumcenter.
In service MI-01-396Parabola Calculator
Find the focus and directrix of a parabola y = ax² from its coefficient, with the equidistance check worked out step by step.
In service MI-01-397Parallel Line Calculator
Find the y-intercept of a line parallel to a given slope that passes through a chosen point, solved from y = mx + b with the working shown.
In service MI-01-398Parallelogram Area Calculator
Enter two adjacent sides and the angle between them to get a parallelogram's area — A = a·b·sinθ, with the hidden height shown along the way.
In service MI-01-399Parallelogram Perimeter Calculator
Enter two adjacent side lengths to get a parallelogram's perimeter — P = 2(a+b), since opposite sides are always equal in length.
In service MI-01-400Partial Fraction Decomposition Calculator
Split N ⁄ ((x−p)(x−q)) into two simpler fractions, A ⁄ (x−p) + B ⁄ (x−q), with two distinct linear factors in the denominator.
In service MI-01-401Partial Products Calculator
Multiplies two two-digit numbers by splitting both into tens and ones, producing all four partial products before summing them into the total.
In service MI-01-402Pascal's Triangle Calculator
Returns the entry at any row and position in Pascal's Triangle via C(n,k), and shows how the same figure falls out of the triangle's own addition rule.
In service MI-01-403Pentagon Calculator
A regular pentagon's area and perimeter from one side length — A = 5s² ⁄ (4·tan 36°), the five-triangle formula behind the Pentagon building's footprint.
In service MI-01-404Percent Error Calculator
How far off was your measurement? Enter measured and actual values; get |measured − actual| ⁄ |actual| × 100 to three decimals, working shown.
In service MI-01-405Percent to Goal Calculator
Turn a current reading and a fixed target into percent of goal reached — current ⁄ goal × 100, uncapped, so passing the target reads honestly above 100%.
In service MI-01-001Percentage Calculator
Three ways round: X% of Y, X as a percent of Y, and percent change between two figures.
In service MI-01-406Percentage Change Calculator
Divides the move in a value by where it started, not where it ended — (new − old) ⁄ old × 100, with the working shown at every step.
In service MI-01-407Percentage Decrease Calculator
Measures how far a value fell as a share of where it started — (old − new) ⁄ old × 100, always reported as a positive percentage.
In service MI-01-408Percentage Difference Calculator
Compare two independent values without picking a 'before' — percentage difference divides by their average, not either value alone.
In service MI-01-409Percentage Increase Calculator
Measures how much larger a value has grown, relative to where it started — (new − old) ⁄ old × 100, always zero or positive, working shown.
In service MI-01-410Percentage Increase Classic
Finds the percent change between an original value and a new value already known — positive for growth, negative for decline, zero for no change at all.
In service MI-01-411Percentage of a Percentage Calculator
Multiply one percentage by another — 50% of 20% is 10%, not 70% — the arithmetic behind nested rates, funnels, and layered shares.
In service MI-01-412Percentage Point Calculator
Finds the plain gap between two rates that are already percentages, points = p1 − p2, with no averaging and no division anywhere.
In service MI-01-413Perfect Cube Calculator
Check whether a whole number is a perfect cube — round its cube root, cube it back, and see if the two figures match exactly.
In service MI-01-414Perfect Square Calculator
Test whether n is a perfect square with an exact round-trip check: round √n to the nearest integer, square it, and compare to n.
In service MI-01-415Perfect Square Trinomial Calculator
Test whether ax² + bx + c is a perfect square trinomial — enter a, b, and c and read the discriminant b² − 4ac instantly.
In service MI-01-416Perimeter Calculator
Pick a basic shape and get its perimeter (or circumference) instantly: circle, square, rectangle, or equilateral triangle, all from one page.
In service MI-01-417Perimeter of a Polygon Calculator
Multiply the number of sides by the length of one side for the exact perimeter of any regular polygon, from a triangle up to a 1000-gon.
In service MI-01-418Perimeter of a Quadrilateral Calculator
Add four side lengths to get a quadrilateral's perimeter — P = a + b + c + d works for any four-sided shape, no matter its angles.
In service MI-01-419Perimeter of a Rectangle Calculator
Add a rectangle's length and width, then double the sum for the total distance around it — P = 2(l + w), worked with your own two side measurements.
In service MI-01-420Perimeter of a Rectangle with Given Area Calculator
Recover a rectangle's hidden width from its area and one known side, then compute the full perimeter for trim, fencing, or edging.
In service MI-01-421Perimeter of a Right Triangle Calculator
Enter both legs of a right triangle and get the full perimeter — the hypotenuse is recovered first, then all three sides are summed.
In service MI-01-422Perimeter of a Sector Calculator
Find the boundary length of a circular wedge: the curved arc plus the two straight radius edges that close it, from radius and central angle.
In service MI-01-423Perimeter of a Square Calculator
Enter one side length and get a square's full perimeter instantly: P = 4s, the quickest possible route from a single measurement to a total boundary length.
In service MI-01-424Perimeter of a Triangle Calculator
Add three side lengths and get a triangle's perimeter instantly — works for scalene, right, obtuse, or equilateral shapes alike.
In service MI-01-425Perimeter of a Triangle with Fractions Calculator
Add three side lengths given as halves, quarters, or eighths and get the exact perimeter — fraction arithmetic done as clean decimal addition.
In service MI-01-426Perimeter of a Triangle with Vertices Calculator
Enter three vertex coordinates and get the triangle's perimeter — the two-point distance formula applied once per side, then summed.
In service MI-01-427Permutation and Combination Calculator
Runs one pair of numbers through both counting formulas at once — P(n,r) for ordered picks, C(n,r) for unordered ones — showing what order is worth.
In service MI-01-428Perpendicular Line Calculator
Find the slope and y-intercept of the line perpendicular to a given line through a specific point, using the negative-reciprocal rule.
In service MI-01-429Phase Shift Calculator
Enter b and c from y = sin(bx + c) to get the phase shift −c ⁄ b — the exact sideways offset from a plain sine wave, in degrees, radians, or turns.
In service MI-01-430Pi Experiments Calculator
Compute any multiple of π directly from Machin's 1706 arctangent formula, without ever looking the constant up.
In service MI-01-431Place Value Calculator
Pull the digit and its place value out of any whole number at a chosen position — 0 for ones, 1 for tens, 2 for hundreds, and upward from there.
In service MI-01-432Point Slope Form Calculator
Enter a slope and one known point and evaluate y = m(x − x1) + y1 at any x — no y-intercept to solve for first, working shown at every step.
In service MI-01-433Polar Coordinates Calculator
Convert a point's x and y coordinates into radius r and angle θ with atan2, so the correct quadrant is recovered automatically from the signs alone.
In service MI-01-434Polar Decomposition Calculator
Split a 2×2 matrix A into a rotation or reflection U and a pure stretch P, then read P's trace and determinant straight from A's singular values.
In service MI-01-435Polar Form Calculator
Convert a complex number's polar description, a modulus and an angle, back into its ordinary rectangular a+bi form.
In service MI-01-436Polar to Cartesian Coordinates Calculator
Convert a point's distance and angle from the origin into Cartesian x and y coordinates, using x = r·cos(θ) and y = r·sin(θ), with every step shown.
In service MI-01-437Polar to Rectangular Coordinates Calculator
Convert polar coordinates (distance and angle) to rectangular (x, y) coordinates: x = r·cos(θ), y = r·sin(θ) — the everyday coordinate-geometry version.
In service MI-01-438Polish Notation Converter
Evaluate a prefix expression like + a * b c, where the operator sits before its operands and nesting alone fixes the order.
In service MI-01-439Polygon Angle Calculator
Give this sheet the number of sides and it returns a regular polygon's interior and exterior angles — (n−2)×180°⁄n and 360°⁄n, always summing to 180°.
In service MI-01-440Polygon Calculator
Enter the number of sides and one side length to get a regular polygon's area and perimeter together, for any n from 3 upward.
In service MI-01-441Polynomial Division Calculator
Divide a quadratic ax² + bx + c by the linear factor (x − r) using synthetic division — get the quotient's coefficients and the remainder in one pass.
In service MI-01-442Polynomial Graphing Calculator
Evaluate a cubic and its own derivative at a chosen point — the curve's height and its tangent-line slope, together, at one x-value.
In service MI-01-443Potato Calculator
Scale potatoes needed for a group by multiplying headcount by an amount per person — the same logic behind sizing rice, punch, or pizza.
In service MI-01-444Power Function Calculator
Evaluate any power function y = a·xᵇ at a chosen x — a fixed exponent applied to a variable base, distinct from an exponential function.
In service MI-01-445Power Mod Calculator
Raise a base to a whole-number exponent and reduce the result by a modulus in one step — aᵇ mod m, worked exactly, no modulus restricted to primes.
In service MI-01-446Power of 10 Calculator
Raise 10 to any whole-number exponent — positive, zero, or negative — and get the exact result, digit by digit.
In service MI-01-447Power of 2 Calculator
Raise 2 to any exponent — positive, negative, or zero — and read 2ⁿ exactly, from a single doubling to a full binary count.
In service MI-01-448Power of a Power Calculator
Raise an already-raised power to another power: (baseᵐ)ⁿ becomes base^(mn), with the multiplication step shown instead of hidden.
In service MI-01-449Power Reducing Calculator
Rewrites sin²θ and cos²θ as (1 ∓ cos2θ) ⁄ 2, trading a squared trig function for a plain cosine of the doubled angle.
In service MI-01-450Power Set Calculator
Enter how many elements are in a set and read the exact size of its power set, 2ⁿ — every subset counted once, from the empty set to the whole thing.
In service MI-01-451Powers of i Calculator
Find any power of the imaginary unit i without repeated multiplication — the powers of i cycle through just four values forever, based on the exponent mod 4.
In service MI-01-452Prime Factorization Calculator
Check a claimed prime factorization by testing whether the product matches and whether each claimed factor genuinely passes a primality test.
In service MI-01-453Prime Number Calculator
Test whether a number is prime by raising a witness base to the power n minus 1 and reducing modulo n — a fast screen, though not a guarantee for every result.
In service MI-01-454Prisoner's Dilemma Calculator
Find both players' payoffs for any combination of cooperate/defect choices, using the classic game-theory payoff matrix.
In service MI-01-455Proportion Calculator
Solve for the missing fourth term in a proportion using cross-multiplication: x = (c × b) ⁄ a, when two ratios are set equal.
In service MI-01-456Pseudoinverse Calculator
Compute the Moore-Penrose pseudoinverse of a 2×2 matrix from a, b, c, and d — identical to the plain inverse whenever the matrix is invertible.
In service MI-01-457Pyramid Angle Calculator
Enter a right pyramid's height and half its base width to get the apex angle, θ = atan(height ⁄ half-base) — the slope of its triangular faces.
In service MI-01-458Pyramid Volume Calculator
Find the volume of a pyramid or cone from its base area and height using V = ⅓ × base area × height, worked out exactly, step by step.
In service MI-01-459Pythagoras Triangle Calculator
Enter three side lengths, longest as c, and get the exact test value a² + b² − c² — zero confirms a right triangle, like 3-4-5 or 5-12-13.
In service MI-01-460Pythagorean Theorem Calculator
Two sides of a right triangle in, the third out: c = √(a² + b²) for the hypotenuse, b = √(c² − a²) for a missing leg. Formula and working shown.
In service MI-01-461Pythagorean Triples Calculator
Pick two whole numbers, m and n, and Euclid's formula returns an exact integer right-triangle triple, with a² + b² = c² guaranteed by construction.
In service MI-01-462QR Decomposition Calculator
Split a 2×2 matrix into an orthogonal matrix Q and an upper-triangular matrix R using the Gram-Schmidt process, reading off R's diagonal.
In service MI-01-463Quadratic Formula Calculator
Solve any quadratic ax² + bx + c = 0 for both roots at once from its three coefficients, with the ± split shown as two separate values.
In service MI-01-464Quadrilateral Calculator
Enter three interior angles of any simple quadrilateral and get the fourth instantly — all four always sum to exactly 360°.
In service MI-01-465Quarter Circle Area Calculator
Find the surface enclosed by a quarter circle from its radius — A = πr²⁄4, exactly one fourth of the full circle's area, worked to the decimal.
In service MI-01-466Quarter Circle Calculator
Enter a quarter circle's radius and get both area and perimeter at once, each formula worked out to full precision from a single figure.
In service MI-01-467Quarter Circle Perimeter Calculator
Find the boundary length of a quarter circle from its radius alone — two straight radius edges plus a quarter-turn arc, P = 2r + πr⁄2.
In service MI-01-468Quaternion Calculator
Multiply two quaternions with the Hamilton product and see why swapping their order can change the answer — unlike real or complex numbers.
In service MI-01-469Queueing Theory Calculator
Find a single-server queue's utilization, average length, and average wait from its arrival and service rates — the classic M/M/1 model.
In service MI-01-470Quiz: Area of a Rectangle Calculator
Work out a rectangle's area in your head, type in your guess, and see the correct answer plus exactly how far off your estimate landed.
In service MI-01-471Quiz: Cylinder Volume Calculator
Guess a cylinder's volume before checking it. Enter radius, height, and an estimate, then see the exact volume and how close the guess came.
In service MI-01-472Quiz: Cylinder Volume Liters Calculator
Guess a cylinder's capacity in liters before checking it — enter radius and height in centimeters, add your estimate, and see exactly how far off you were.
In service MI-01-473Quiz: Right Triangle Side and Angle Calculator
Estimate the angle opposite a known leg before checking — enter a leg and hypotenuse, guess the angle in degrees, and see exactly how close you came.
In service MI-01-474Quiz: Trigonometry Calculator
Recall sin of an angle from memory before checking — enter the angle, type your estimate, and see the exact value plus how close you came.
In service MI-01-475Quotient Calculator
Divide a dividend by a divisor and read the exact decimal quotient, computed to full precision with no remainder left dangling.
In service MI-01-476Radical Calculator
Compute √(a²b) two separate ways at once — directly, and as a√b after pulling a perfect square outside — to watch the simplification identity hold.
In service MI-01-477Radius of a Circle Calculator
Solve A = πr² backwards: enter a circle's area and get its exact radius, the specific inverse this sheet is built for.
In service MI-01-478Radius of a Cone Calculator
Recover a cone's base radius from a known volume and perpendicular height by inverting V = ⅓πr²h, the cone volume identity, for r.
In service MI-01-479Radius of a Cylinder Calculator
Rearrange a cylinder's volume formula to solve for its radius when the capacity and height are already known: r = √(V ⁄ πh).
In service MI-01-480Radius of a Sphere Calculator
Recover a sphere's radius from its volume by inverting V = (4⁄3)πr³ — a cube root, not a square root, because volume scales with radius cubed.
In service MI-01-481Ratio Calculator
Reduce a ratio to its lowest terms by dividing both numbers by their greatest common divisor — the same logic used to simplify a fraction.
In service MI-01-482Ratio of 2 Numbers Calculator
Reduce any ratio a:b to its lowest whole-number terms by dividing both numbers by their greatest common divisor, GCD shown.
In service MI-01-483Ratio of 3 Numbers Calculator
Reduce a three-number ratio to lowest terms by chaining two GCD calls into one shared divisor, with every step of the simplification shown.
In service MI-01-484Rational Exponents Calculator
Confirm the rational-exponent identity x^(m⁄n) = (xᵐ)^(1⁄n) = (x^(1⁄n))ᵐ by computing both orders and checking they agree.
In service MI-01-485Rational Zeros Calculator
Test whether a candidate integer is even a plausible root of a monic polynomial, using the Rational Root Theorem's divisibility rule.
In service MI-01-486Rationalize Denominator Calculator
Clears the square root out of a fraction's denominator, turning a⁄√b into a√b⁄b, with the multiply-by-√b step shown in full.
In service MI-01-487Ratios of Directed Line Segments Calculator
Split a segment between two points in any ratio m:n and land on the exact dividing point — the section formula, worked out on both axes.
In service MI-01-488Reciprocal Calculator
Find the multiplicative inverse of any nonzero number: 1 ⁄ x — the value that, multiplied by the original, always gives exactly 1.
In service MI-01-489Rectangle Diagonal Angle Calculator
Find the angle a rectangle's diagonal makes with its width from θ = atan(height ⁄ width) — the same right triangle hiding in every corner.
In service MI-01-490Rectangle Scale Factor Calculator
Divide a resized dimension by the original one to get the exact scale factor, and see why area grows by its square, not by that number alone.
In service MI-01-491Rectangular Prism Calculator
Find a rectangular prism's (box's) volume and total surface area from its three dimensions: V = lwh and A = 2(lw + lh + wh).
In service MI-01-492Rectangular Pyramid Volume Calculator
Multiply length, width, and height by one-third to find a rectangular pyramid's volume, with the full working shown.
In service MI-01-493Rectangular to Polar Coordinates Calculator
Convert plain (x, y) coordinates to polar form — distance and angle: r = √(x²+y²), θ = atan2(y, x).
In service MI-01-494Reference Angle Calculator
Find the acute angle between any angle's terminal side and the x-axis — one formula covers negative angles and angles past 360° too.
In service MI-01-495Relative Change Calculator
Track how much a value has grown or declined, as a signed percentage: (new − old) ⁄ old × 100 — positive for growth, negative for decline.
In service MI-01-496Relatively Prime Calculator
Check whether two whole numbers are relatively prime by computing gcd(a, b) directly — an exact 1 means the pair is coprime.
In service MI-01-497Remainder Calculator
Find what's left over after dividing one number by another — the same leftover long division produces as its final step.
In service MI-01-498Reverse FOIL Calculator
Expand (x+p)(x+q) forward into its trinomial coefficients — the direction that makes reversing FOIL to factor click.
In service MI-01-499Rhombus Area Calculator
Enter a rhombus's two diagonals to get its exact area — A = ½·d₁·d₂, valid because the diagonals always cross at a right angle and bisect each other.
In service MI-01-500Right Circular Cone Calc: find A, V, A_L, A_B.
Enter a cone's radius and height once and get slant height, volume, lateral area, and total surface area together, all four figures cross-checked.
In service MI-01-501Right Cylinder Calculator: find A, V, A_l, A_b
Solve a right cylinder's volume, lateral (side) area, and total surface area together from just the radius and height.
In service MI-01-502Right Rectangular Pyramid Calc: find A, V, A_l, A_b
Enter a rectangular base's length and width plus a height to get volume, base area, lateral area, and total surface area together.
In service MI-01-503Right Square Pyramid Calc: find A, A_l, V, A_F
Enter a right square pyramid's base side and height to get slant height, volume, lateral area, and total surface area together.
In service MI-01-504Right Trapezoid Area Calculator
Give a right trapezoid's shorter base, height, and slant leg — recover the missing longer base by the Pythagorean theorem, then read the area.
In service MI-01-505Right Trapezoid Calculator
Give a right trapezoid's two bases and its perpendicular height — recover the slant leg by the Pythagorean theorem, then read area and perimeter.
In service MI-01-506Right Triangle Calculator
Enter a right triangle's two legs and get the hypotenuse, area, perimeter, and angle opposite Leg a — all four solved together, no repeat entry.
In service MI-01-507Right Triangle Side and Angle Calculator
Solve a right triangle from one leg and the hypotenuse — the other leg and both acute angles, all recovered at once.
In service MI-01-508Right Triangle Trigonometry Calculator
Enter a measured angle plus the hypotenuse to get the opposite and adjacent legs of a right triangle, straight from sine and cosine.
In service MI-01-509Rise Over Run Calculator
Turn a ramp's height and length, or a roof's rise and span, directly into a slope ratio — slope = rise ⁄ run, no coordinate points needed.
In service MI-01-510Root Calculator
Find the nth root of any real number for any whole-number index n, with the even-root domain rule and the odd-root sign rule spelled out exactly.
In service MI-01-511Root Mean Square Calculator
Squares three values first, averages those squares, then roots the result — RMS reports a set's typical size without positive and negative terms cancelling out.
In service MI-01-512Rotation Calculator
Rotate a point (x, y) counterclockwise about the origin by any angle θ, using the 2×2 rotation matrix, in degrees, radians, or turns.
In service MI-01-513Round to the Nearest Cent Calculator
Scales a dollar figure to whole cents, rounds it, and scales back down — round(x × 100) ⁄ 100 shown step by step, with the half-up tie rule made explicit.
In service MI-01-514Round to the Nearest Dollar Calculator
Rounds any dollars-and-cents figure to the nearest whole dollar with round(x) — the same half-up rule the IRS permits for whole-dollar tax returns.
In service MI-01-515Round to the Nearest Hundred Calculator
Divide by 100, round to the nearest whole number, then multiply back by 100 — the exact three-step rule for rounding to the nearest hundred, worked in full.
In service MI-01-516Round to the Nearest Hundredth Calculator
Multiply by 100, round to a whole number, divide back — round(x × 100) ⁄ 100 turns any figure into two clean decimal places, shown step by step.
In service MI-01-517Round to the Nearest Integer Calculator
Round any decimal to the nearest whole number with round(x), the base rounding step every place-value rounding rule is built on top of.
In service MI-01-518Round to the Nearest Penny Calculator
Turn a fractional-cent amount from a tax calc or a split bill into the nearest real cent that can actually be paid — round(x × 100) ⁄ 100, framed for money.
In service MI-01-519Round to the Nearest Ten Calculator
Round any number to the nearest multiple of ten with round(x ⁄ 10) × 10 — the scale-round-rescale method shown step by step, including tie cases.
In service MI-01-520Round to the Nearest Tenth Calculator
Enter any number and get it rounded to one decimal place, with the scale-round-rescale steps and the half-up tie rule shown in full.
In service MI-01-521Round to the Nearest Thousand Calculator
Divide by 1000, round to the nearest whole number, multiply back — any figure rounded to its nearest thousand, with every step of the arithmetic shown.
In service MI-01-522Round to the Nearest Thousandth Calculator
Scales a decimal by 1000, rounds to a whole number, and scales back down — round(x × 1000) ⁄ 1000 with the tie rule shown step by step.
In service MI-01-523Rounding Calculator
Multiply by ten raised to the decimal-place count, round to a whole number, and divide back — round(x × 10ᵈ) ⁄ 10ᵈ worked for any precision from 0 to 10 places.
In service MI-01-524RSA Calculator
Turn two primes into a public modulus and totient, then encrypt a message with RSA's c = mᵉ mod n — every step of textbook RSA shown.
In service MI-01-525SAS Triangle Calculator
Enter two sides and the angle between them; this sheet solves the whole triangle — third side, both remaining angles, and area.
In service MI-01-526Scalene Triangle Area Calculator
Compute a scalene triangle's area from its three side lengths using Heron's formula — enter a, b, and c and read the area directly.
In service MI-01-527Scalene Triangle Calculator
Find a scalene triangle's area from its three sides via Heron's Formula, then confirm it's genuinely scalene with a built-in distinctness test.
In service MI-01-528Scatter Plot Calculator
Finds the slope of the line joining two data points — the simplest possible scatter plot and the building block behind fitting a trend across many.
In service MI-01-529Scientific Notation Calculator
Convert any number to scientific notation: a mantissa between 1 and 10, multiplied by a power of 10 — exponent = ⌊log₁₀|x|⌋.
In service MI-01-530Scientific Notation Converter
Turn any nonzero number into scientific notation: a mantissa between 1 and 10, multiplied by the right power of 10.
In service MI-01-531Scientific Notation Equation Calculator
Multiply two numbers already written in scientific notation, with the mantissa automatically renormalized back under 10 when the product overflows it.
In service MI-01-532Secant Calculator
Flip cosine to get secant, watch it rocket toward infinity at 90° and 270°, and see why its value can never land between −1 and 1.
In service MI-01-533Sector Area Calculator
Find the surface enclosed by a circular wedge from its radius and central angle — A = ½r²θ, the fraction of the whole circle it claims.
In service MI-01-534Segment Addition Postulate Calculator
Add two adjacent segment lengths, AB and BC, to find the whole distance AC — the postulate stating that a segment's parts sum to its total.
In service MI-01-535Segment Area Calculator
Find the area of a circular segment, the sliver a chord cuts off a circle, by taking sector area and subtracting the triangle beneath it.
In service MI-01-536Semicircle Area Calculator
Enter a radius and get a semicircle's area from A = πr²⁄2 — exactly half the parent circle's area, worked to the decimal every time.
In service MI-01-537Sequence Calculator
Find the nth term of an arithmetic or geometric sequence from its first term and common difference or ratio, with a type selector.
In service MI-01-538Set Builder Calculator
Counts how many whole numbers in a range are divisible by a target, framed with set-builder notation: {n in [a,b] : n divisible by k}.
In service MI-01-539Significant Figures Calculator – Sig Fig
Round any number to a chosen count of significant figures, showing the order of magnitude and rounding scale behind each result.
In service MI-01-540Similar Right Triangles Calculator
Drop a right triangle's own altitude onto its hypotenuse and the two segments p and q set the altitude exactly: altitude = √(p × q).
In service MI-01-541Similar Triangles Calculator
Give one matched pair of sides from two similar triangles and this sheet returns the scale factor, then applies it to scale any other side.
In service MI-01-542Simplify Cube Root Calculator
Verify the cube-root simplification identity by computing both sides directly: pulling a perfect-cube factor out from under a cube root.
In service MI-01-543Simplify Fractions Calculator
Divide a fraction's numerator and denominator by their greatest common divisor to reduce it to lowest terms, with the GCD shown.
In service MI-01-544Simplify Ratio Calculator
Reduce any ratio a:b to its simplest whole-number form by dividing both sides by their greatest common divisor.
In service MI-01-545Simplifying Radicals Calculator
Clear a radical out of a fraction's denominator by multiplying top and bottom by that same radical, verified by computing both forms directly.
In service MI-01-546Sin 2 Theta Calculator
Multiply an angle's own sine and cosine, double the product, and get sin(2θ) — the identity behind projectile range and the sine curve's steepest climb.
In service MI-01-547Sin Degrees Calculator
Find the y-coordinate of a point rotating around a unit circle at a given angle — sine, seen directly as a coordinate.
In service MI-01-548Sin Theta Calculator
Find a wave's instantaneous height at a given phase angle, scaling the plain sine curve by a chosen amplitude.
In service MI-01-549Sin-1 Calculator
What pressing sin⁻¹ on a scientific calculator returns, why it differs from 1 ÷ sin(x), and how the button matches asin(x) and arcsin(x) exactly.
In service MI-01-550Sine Calculator
Enter an angle in degrees, radians, or turns and read off sin(θ) — the height of that angle's point on a unit circle, computed exactly.
In service MI-01-551Sine Cosine Tangent Calculator
Enter one angle and read sin(θ), cos(θ), and tan(θ) together — the three-function companion to the separate sine, cosine, and tangent sheets.
In service MI-01-552Sine Function Calculator
Evaluate the sine of any angle: sin(x) — the y-coordinate of the matching point on the unit circle, oscillating smoothly between −1 and 1.
In service MI-01-553Sine Triangle Calculator
Find a right triangle's opposite side directly from its hypotenuse and angle — a single-question convenience version of the full solver.
In service MI-01-554Singular Values Calculator
Find a 2×2 matrix's two singular values, the amounts it stretches space along its most- and least-amplified directions.
In service MI-01-555Sinh Calculator
Evaluate sinh(x) = (eˣ − e⁻ˣ) ⁄ 2 instantly — the odd, unbounded hyperbolic sine that gives a catenary its slope and momentum its rapidity term.
In service MI-01-556Slant Height calculator
Find a cone's slant height from its base radius and perpendicular height: slant = √(r² + h²), the Pythagorean theorem applied to the cone's own cross-section.
In service MI-01-557Slant Height of a Cone Calculator
Combine a cone's radius with its height using l = √(r² + h²) to find the exact length of the sloped edge, worked step by step.
In service MI-01-558Slope Calculator
Enter two points and read the slope as rise over run, plus the line's angle of inclination in degrees via the arctangent. Working shown.
In service MI-01-559Slope Intercept Form Calculator
Enter a slope and y-intercept, plug in any x, and read y = mx + b instantly — the most recognizable way to write a line's equation, worked step by step.
In service MI-01-560Slope Percentage Calculator
Turn a rise and a run measurement into the grade percentage used on road signs, ADA ramp codes, and topographic maps, no protractor required.
In service MI-01-561Solve Similar Triangles Calculator
Recover a similar triangle's two missing sides from one full reference triangle and just one corresponding side of the second.
In service MI-01-562Solving Quadratic Equations by Completing the Square
Solve ax² + bx + c = 0 by completing the square, returning p and q where (x+p)² = q — the visual, step-revealing alternative to the quadratic formula.
In service MI-01-563Special Right Triangles Calculator
Find every side of a 30-60-90 triangle from just the short leg: sides always sit in the fixed ratio 1 : √3 : 2.
In service MI-01-564Sphere Calc: find V, A, d
Enter one radius and get a sphere's volume, surface area, and diameter together — three formulas solved from a single measurement.
In service MI-01-565Sphere Volume Calculator
Find a sphere's volume from its radius: V = (4⁄3)πr³ — the single figure returned directly, without a combined surface-area solver's extra output.
In service MI-01-566Spherical Coordinates Calculator
Convert a radial distance and two angles, azimuth and polar, into Cartesian x, y, and z using x = r·sinφ·cosθ, worked out step by step.
In service MI-01-567Spiral Length Calculator
Find the arc length of an Archimedean spiral, r = a·θ, from its center to any swept angle — the curve behind vinyl grooves and rolled hoses.
In service MI-01-568Square Calc: find A, P, d
Enter one side length and get a square's area, perimeter, and diagonal together in a single pass — three formulas checked against one measurement.
In service MI-01-569Square feet of a Rectangle Calculator
Multiply length by width to get a rectangle's area in square feet — the figure flooring, tile, carpet, and paint quotes are priced against.
In service MI-01-570Square Feet of a Triangle Calculator
Enter a triangular section's base and height in feet and read the area straight in square feet — the ½·base·height formula, sized for materials orders.
In service MI-01-571Square Footage of a Circle Calculator
Enter a circle's radius in feet and read its area in square feet from A = πr² — the figure landscapers, pavers, and pool installers order material against.
In service MI-01-572Square in a Circle Calculator
Enter a circle's radius to find the largest square that fits inside it — the inscribed side length and area, both derived from r times root 2.
In service MI-01-573Square Inches of a Circle Calculator
Turn a radius into a circle's area in square inches — the working unit for pins, gaskets, drilled holes, and pressure math, not room-sized floors.
In service MI-01-574Square of a Binomial Calculator
Expand (ax + b)² into a², 2ab, and b² — the trinomial coefficients a squared binomial always produces, middle term included.
In service MI-01-575Square Pyramid Calculator
Find a square pyramid's volume and total surface area from its base edge and height, using the slant height along the way.
In service MI-01-576Square Pyramid Volume Calculator
Find a square pyramid's volume from its base edge and height: V = s²h ⁄ 3 — just the one figure, no extra surface-area output.
In service MI-01-577Square Root Calculator
Find √x for any number zero or above, with the domain rule, the principal-root convention, and a fully worked example shown exactly.
In service MI-01-578SSA Triangle Calculator
Enter two sides and a non-included angle to see whether zero, one, or two triangles fit the numbers, and why SSA alone stays ambiguous.
In service MI-01-579SSS Triangle Calculator
Solve a triangle completely from its three side lengths alone — all three angles recovered by applying the Law of Cosines three times over.
In service MI-01-580Standard Equation of a Circle Calculator
Enter three points that lie on a circle and recover its center (h,k) and radius r, reconstructed entirely from those points via two perpendicular bisectors.
In service MI-01-581Standard Form Calculator
Enter two points and get the line's standard form, Ax + By = C, with B fixed at −1 and every coefficient shown exactly.
In service MI-01-582Standard Form to General Form of a Circle Calculator
Expand a circle's standard-form equation, (x−h)²+(y−k)²=r², into general form, x²+y²+Dx+Ey+F=0: D=−2h, E=−2k, F=h²+k²−r².
In service MI-01-583Standard Form to Slope Intercept Form Calculator
Convert Ax + By = C into y = mx + b by isolating y — enter A, B, and C and read off the slope and y-intercept at once.
In service MI-01-584Standard Notation Calculator
Expand a mantissa-and-exponent pair, m × 10ⁿ, back into the plain, comma-ready number it stands for — the reverse of scientific notation.
In service MI-01-585Star Shape Calculator
Give any four point angles of a simple five-pointed star and this sheet finds the fifth — the five tips always add up to exactly 180°.
In service MI-01-586String Girdling Earth Calculator
Splice extra length into a loop around any circle — a ring, a barrel, or Earth's equator — and get the uniform gap it opens: gap = ΔC ⁄ 2π.
In service MI-01-587Subset Calculator
Count every subset a set of n elements can form, using 2ⁿ — the product of two independent choices, made separately for each element in turn.
In service MI-01-588Substitution Method Calculator
Find where a straight line crosses a parabola by substituting the line's own expression in for y, then solving the resulting quadratic directly.
In service MI-01-589Subtracting Fractions Calculator
Cross-multiply two fractions onto a shared denominator and subtract, returning the exact result numerator and denominator, unreduced and unrounded.
In service MI-01-590Subtraction Calculator
Subtract one number from another and get the exact difference, including a negative result when the second number is the larger one.
In service MI-01-591Sum and Difference Identities Calculator
Expand sin(A+B) and cos(A+B) from the separate sines and cosines of two angles, and read off the difference forms by flipping one sign.
In service MI-01-592Sum of a Linear Number Sequence Calculator
Enter a first term, common difference, and number of terms to get the exact total of an arithmetic run, without adding every term by hand.
In service MI-01-593Sum of Products Calculator
Enter two numbers and read their sum and product together — the forward half of the relationship the diamond-problem solver runs in reverse.
In service MI-01-594Sum of Series Calculator
Add up a geometric series from its first term, common ratio, and term count — S = a₁(1 − rⁿ) ⁄ (1 − r), worked out exactly.
In service MI-01-595Supplementary Angles Calculator
Find the angle that completes a straight line with yours: supplement = 180° − θ, with every step of the subtraction shown in full.
In service MI-01-596Surface Area Calculator
Pick a basic solid and get its total surface area instantly: cube, sphere, cylinder, or cone, all from one page with a shape selector.
In service MI-01-597Surface Area of a Cone Calculator
Find a cone's slant height, lateral surface area, and total surface area from its base radius and height, with every step shown.
In service MI-01-598Surface Area of a Cube Calculator
Find the total surface area of a cube from its side length — SA = 6s², six identical square faces summed and shown step by step.
In service MI-01-599Surface Area of a Cylinder Calculator
Add the curved wall to both circular caps for a cylinder's complete outside surface, A = 2πrh + 2πr², the figure a full paint or plating job needs.
In service MI-01-600Surface Area of a Hemisphere Calculator
Enter a hemisphere's radius to get its curved surface area (2πr²) and its total surface area with the flat base included (3πr²).
In service MI-01-601Surface Area of a Rectangle Calculator
Double a rectangle's length-times-width area to get the total surface of both faces — the figure real sheet material needs covered.
In service MI-01-602Surface Area of a Rectangular Prism Calculator
Find a box's total surface area directly from its length, width, and height — a quick single-figure convenience page.
In service MI-01-603Surface Area of a Rectangular Pyramid Calculator
Find a rectangular-based pyramid's total surface area — TWO different slant heights needed, one for each pair of triangular side faces.
In service MI-01-604Surface Area of a Square Pyramid Calculator
Find a square pyramid's total surface area from its base edge and height — one shared slant height covering all four identical triangular faces.
In service MI-01-605Surface Area of a Triangular Prism Calculator
Enter a triangle's three sides and a prism's length to get total surface area — Heron's formula for the caps, perimeter times length for the walls.
In service MI-01-606Surface Area to Volume Ratio Calculator
Divide a solid's surface area by its volume to get the SA:V ratio that governs heat loss, diffusion, and how a shape trades with its surroundings.
In service MI-01-607SVD Calculator
Find the 2-norm condition number of a 2×2 matrix, the ratio of its largest to smallest singular value, a standard gauge of numerical sensitivity.
In service MI-01-608Synthetic Division Calculator
Divide a cubic polynomial by a linear factor (x−r) using synthetic division — a compact shortcut for polynomial long division.
In service MI-01-609System of Equations Calculator
Solve a 2-by-2 system of linear equations directly using Cramer's Rule, two determinant-ratio formulas that skip substitution or elimination.
In service MI-01-610Tan Inverse Calculator
Find an incline angle directly from a rise-over-run slope ratio using inverse tangent, a ramp-and-grade framing of arctangent.
In service MI-01-611Tan-1 Calculator
What the tan⁻¹ button on a scientific calculator returns, why any real number is a valid entry, and how it matches atan(x) and arctan(x) exactly.
In service MI-01-612Tangent Angle Calculator
Find the tangent of any angle directly — the forward direction of inverse tangent, for plugging into other formulas.
In service MI-01-613Tangent Calculator
Enter an angle and get its tangent, computed as sin(θ) divided by cos(θ), including the vertical asymptotes where it blows up.
In service MI-01-614Tangent of a Circle Calculator
Find how long a tangent line is from a point outside a circle to the point where it touches, using length = √(d² − r²).
In service MI-01-615Tangent Ratio Calculator
Evaluate the tangent of any angle: tan(x) = sin(x) ⁄ cos(x) — a right triangle's opposite side over its adjacent side, for that angle.
In service MI-01-616Tanh Calculator
Evaluate tanh(x) = sinh(x) ⁄ cosh(x) instantly — the odd, bounded ratio that squeezes every real input into the open band between −1 and 1.
In service MI-01-617Tensor Product Calculator
Multiply every pair of components from two vectors and keep all four results separate, arranged as a 2×2 matrix with nothing summed away.
In service MI-01-618Terminating Decimals Calculator
Test whether a fraction's decimal form terminates or repeats forever, using a single closed-form modulo test on its reduced denominator.
In service MI-01-619Tetrahedron Volume Calculator
Enter one edge length and this sheet returns a regular tetrahedron's volume from V = a³ ⁄ (6√2), with every step of the working shown.
In service MI-01-620Theta Calculator
Find the reference angle for any angle, including negatives or values past 360°: the equivalent acute angle between 0° and 90°.
In service MI-01-621Three-Dimensional Distance Calculator
Find the straight-line separation between two points in 3D space by extending the Pythagorean theorem through a third axis.
In service MI-01-622Time Percentage Calculator
Find what percentage of a total time block has been used: time spent ⁄ total time × 100 — for billable hours, task tracking, or screen time.
In service MI-01-623Torus Surface Area Calculator
Find a torus's full outer surface area from its major and minor radii using A = 4π²Rr, with the two-circumference sweep behind it explained.
In service MI-01-624Torus Volume Calculator
Find how much a donut-shaped ring holds from its major and minor radii using V = 2π²Rr², built on Pappus's theorem for solids of revolution.
In service MI-01-625Trapezoid Angle Calculator
Find a trapezoid's base angle from its slant leg and height using asin(height ⁄ leg) — the right-triangle relationship hiding inside every non-parallel side.
In service MI-01-626Trapezoid Calculator
Enter both bases, the height, and both legs to get a trapezoid's area and perimeter together from one set of measurements.
In service MI-01-627Trapezoid Height Calculator
Recover a trapezoid's height from its area and two parallel base lengths — h = 2A ⁄ (b₁+b₂), the area formula solved for h instead of A.
In service MI-01-628Trapezoid Perimeter Calculator
Add a trapezoid's two bases and two legs to find its perimeter — P = b1+b2+leg1+leg2, with no height or area needed.
In service MI-01-629Trapezoid Side Calculator
Recover a general trapezoid's missing leg from both bases, the known leg, and the height — no assumption that the two legs are equal.
In service MI-01-630Triangle Angle Calculator
Find a triangle's third interior angle from its other two, using the fact that every triangle's angles sum to exactly 180°.
In service MI-01-631Triangle Area Calculator
Enter the base and the perpendicular height, read the area — in cm, m, or ft — with the half-rectangle formula A = b·h/2 shown and explained.
In service MI-01-632Triangle Congruence Calculator
Enter both triangles' three sides in any order and get a single SSS congruence test: zero means identical, anything else measures the mismatch.
In service MI-01-633Triangle Degree Calculator
Convert a triangle's interior angle from radians to degrees, the more intuitive unit for reading a shape's angles directly.
In service MI-01-634Triangle Height Calculator
Recover a triangle's perpendicular height from its known area and base — h = 2A ⁄ base, worked step by step with the check built in.
In service MI-01-635Triangle Inequality Theorem Calculator
Type in three candidate side lengths and see whether they can form a genuine triangle, plus the exact slack or shortfall on the tightest pairing.
In service MI-01-636Triangle Length Calculator
Already know a triangle's perimeter and two of its three side lengths? Subtract both from the total to recover the missing third length instantly.
In service MI-01-637Triangle Proportionality Theorem Calculator
Enter three segments cut by a line parallel to a triangle's third side and this sheet solves for the missing fourth segment, ratio shown.
In service MI-01-638Triangle Ratio Calculator
Divide a triangle's longest side by its shortest to measure how stretched or elongated its shape is, independent of overall size.
In service MI-01-639Triangle Scale Factor Calculator
Find the scale factor between two similar triangles from one matching pair of corresponding sides: scale factor = a₂ ⁄ a₁.
In service MI-01-640Triangle Side Angle Calculator
Solve a right triangle from one angle and its opposite leg, recovering the hypotenuse and the adjacent side in one step.
In service MI-01-641Triangle Side Calculator
Solve a triangle from two angles and the length sandwiched between them — enter Angle A, Angle B, and side c for the missing angle and both remaining sides.
In service MI-01-642Triangle Similarity Calculator
Compare one known side from each of two similar triangles to find the scale factor between them, and how their areas differ.
In service MI-01-643Triangle Slope Calculator
Find the slope of all three edges of a triangle from its plotted corner coordinates — each side's own steepness, computed independently.
In service MI-01-644Triangle Sum Theorem Calculator
Enter any two interior angles of a triangle and recover the third using the Triangle Sum Theorem: every triangle's three angles always add to exactly 180°.
In service MI-01-645Triangle Vertices Calculator
Find a triangle's area and perimeter directly from its three corner coordinates, using the shoelace formula and the Distance Formula three times over.
In service MI-01-646Triangular Numbers Calculator
Find the nth triangular number, the count of dots needed to stack a triangle n rows deep, using Gauss's own shortcut: Tₙ = n(n+1) ⁄ 2.
In service MI-01-647Triangular Prism Calculator
Solve a right-triangle-cross-section prism completely: volume and total surface area together, from the two legs and the prism's length.
In service MI-01-648Triangular Pyramid Volume Calculator
Find the volume of any triangular pyramid (tetrahedron), regular or irregular, from its base area and height: V = base area × height ⁄ 3.
In service MI-01-649Triangulation Calculator
Find the distance to a far-off point from a measured baseline and two sighted angles — classic surveying triangulation via the Law of Sines.
In service MI-01-650Trig Calculator
Enter any angle you like and read off sine, cosine, and tangent together in a single lookup, each given as a decimal.
In service MI-01-651Trig Degree Calculator
Convert an angle from degrees into radians, the unit trigonometry and calculus lean on almost everywhere else.
In service MI-01-652Trig Function Calculator
Choose sine, cosine, or tangent from a dropdown, enter one angle, and get back exactly that function's value, nothing extra computed alongside it.
In service MI-01-653Trig Identities Calculator
Pick any angle and see the Pythagorean trig identity confirmed live: sin²θ and cos²θ always add to exactly 1, for every angle without exception.
In service MI-01-654Trig Triangle Calculator
Give both legs of a right triangle and get the angle between hypotenuse and adjacent leg, plus the hypotenuse itself, via arctangent and Pythagoras.
In service MI-01-655Trigonometric Functions Calculator
Evaluate sin, cos, and tan for the same angle at once — SOH-CAH-TOA's three ratios side by side.
In service MI-01-656Trigonometry Calculator
Enter an angle and get all three reciprocal trig functions at once: secant (1 ⁄ cos), cosecant (1 ⁄ sin), and cotangent (1 ⁄ tan).
In service MI-01-657Truncated Cone Calculator
Solve a frustum completely: volume and total surface area together, from both radii and the height.
In service MI-01-658Truncated Cone Volume Calculator
Find the volume of a truncated cone (frustum) from its two radii and height: V = πh(R² + Rr + r²) ⁄ 3, no intermediate full-cone subtraction needed.
In service MI-01-659Truth Table Generator
Compute AND, OR, and XOR for two 0/1 logic inputs at once, using plain arithmetic identities in place of logic gates.
In service MI-01-660Two's Complement Calculator
Find a 4-bit number's two's complement, the standard binary encoding for negative numbers, computed directly as 16 minus the number.
In service MI-01-661Ugly Duckling Theorem Calculator
Computes how many size-k predicate subsets contain two specific objects at once — always C(n−2, k−2), identical for every pair, per Watanabe's theorem.
In service MI-01-662Union and Intersection Calculator
Find the size of the union of two overlapping sets from their individual sizes and their intersection: |A ∪ B| = |A| + |B| − |A ∩ B|.
In service MI-01-663Unit Circle Calculator
Enter any angle and read off its exact coordinates on the unit circle: x = cos θ, y = sin θ — the definition every other trig identity builds on.
In service MI-01-664Unit Rate Calculator
Turn a total cost for a quantity into a plain per-unit rate: rate = total ⁄ quantity — the number that lets differently sized options be compared fairly.
In service MI-01-665Unit Vector Calculator
Rescale any 2D vector to a length of exactly 1 while keeping its direction unchanged — divide each component by the vector's own magnitude.
In service MI-01-666Vector Addition Calculator
Add two 2D vectors component by component: (x₁, y₁) + (x₂, y₂) = (x₁+x₂, y₁+y₂) — the algebraic version of the tip-to-tail rule.
In service MI-01-667Vector Calculator
Find a vector's magnitude and direction from its components at once: |v| = √(x²+y²), angle = atan2(y,x) — both halves of its description together.
In service MI-01-668Vector Magnitude Calculator
Find a 2D vector's length from its components using the Pythagorean theorem: |v| = √(x² + y²) — the straight-line distance from the origin to the vector's tip.
In service MI-01-669Vector Projection Calculator
Find how much of one vector points along another's direction: proj = (v · u) ⁄ |u| — the dot product turned into a signed length.
In service MI-01-670Vertex Form Calculator
Convert a quadratic from standard form (y = ax² + bx + c) directly to its vertex coordinates (h, k): h = −b ⁄ 2a and k = c − b² ⁄ 4a.
In service MI-01-671Volume Calculator
Pick a basic solid and get its volume instantly: cube, sphere, cylinder, or cone, all from one page with a shape selector.
In service MI-01-672Volume of a Cube Calculator
Cube a side length to get exact volume — V = s³, the formula that gave the operation 'cubing' its name, with every step shown.
In service MI-01-673Volume of a Hemisphere Calculator
Find the volume of a hemisphere from its radius: V = (2 ⁄ 3)πr³ — exactly half of a full sphere's (4 ⁄ 3)πr³.
In service MI-01-674Volume of a Hexagonal Pyramid Calculator
Find the volume of a pyramid with a regular hexagonal base from its side length and height: V = (3√3 ⁄ 2)s² × height ⁄ 3.
In service MI-01-675Volume of a Parallelepiped Calculator
Find the volume of a slanted 3D box from three edge vectors using the scalar triple product: V = |a · (b × c)|.
In service MI-01-676Volume of a Rectangular Prism Calculator
Find a box's volume directly from its length, width, and height — a quick single-figure convenience page.
In service MI-01-677Volume of a Trapezoidal Prism Calculator
Enter a trapezoid's two parallel sides, its height, and a prism length to get volume from cross-sectional area times length.
In service MI-01-678Volume of a Triangular Prism Calculator
Find a triangular prism's volume from its triangular cross-section (base and height) and its own length: V = ½ × base × height × length.
In service MI-01-679Weighted Average Calculator
Average three values that don't count equally, using each one's own weight: weighted average = Σ(value × weight) ⁄ Σweight.
In service MI-01-680Width of a Rectangle Calculator
Recover a rectangle's missing width from its area and length: width = area ⁄ length — the direct algebraic reverse of area = length × width.
In service MI-01-681XOR Calculator
Compute the exclusive-or of two binary inputs: A XOR B = A + B − 2AB — true exactly when the two inputs differ from each other.
In service MI-01-682y Intercept Calculator
Find where a line through two points crosses the y-axis: b = y₁ − m·x₁, using the slope m = (y₂ − y₁) ⁄ (x₂ − x₁) computed first.
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