How this instrument works
A right triangle carries three sides and three angles, but only two of those six numbers need to be independent, since the shape is fixed the moment one angle and one length are set. Here, the two starting measurements are that angle and the length sitting directly across from it — call it the far reading, since it never touches the angle's own vertex. Dividing that reading by the sine of the angle recovers the hypotenuse, and dividing it by the tangent recovers the remaining distance, the one touching the vertex instead.
This site carries several right-triangle solvers, and each begins from a different starting pair. One begins from a known side and the hypotenuse together, using the Pythagorean theorem to find the missing length. Another begins from an angle paired with that same hypotenuse, rather than a side. A third begins from the triangle's two shorter lengths, with no angle supplied at all. Pairing an angle with its own far length instead is a fourth combination, worth reaching for whenever that measurement is the one you can actually put a tape measure on — a flagpole's own height, say, read against the sun's elevation, without ever touching the shadow it casts.
Push that angle toward its extremes and the shape behaves exactly as geometry demands. Near 90°, the adjacent length shrinks toward zero while the hypotenuse converges on the far reading's own size, since a figure with almost no room left between its two acute corners is close to folding flat. Near 0°, by contrast, the hypotenuse balloons outward while the adjacent distance chases after it — the far reading has to stretch further and further to still reach the last vertex.
- Enter the length of the leg you can measure directly into the Opposite side field.
- Enter the known angle into the Angle field, in degrees by default.
- Read Hypotenuse for the longest side, solved from opposite divided by sin(angle).
- Read Adjacent side for the remaining leg, solved from opposite divided by tan(angle).
Worked example — a 5-unit leg at 30°
Take the golden case: an opposite measurement of 5 against a 30° angle. The hypotenuse is 5 divided by sin(30°), which is 5 divided by 0.5, giving 10 exactly, and the adjacent length is 5 divided by tan(30°), roughly 5 divided by 0.57735, landing near 8.660254 — the familiar 30-60-90 triangle, scaled so its shortest measurement reads 5.
Carry a different opposite reading, 3, through 45° instead: the hypotenuse becomes 3 divided by sin(45°), about 3 divided by 0.7071068, or roughly 4.242641 — equal to 3√2 — and the adjacent value comes out to 3 divided by tan(45°), which is 3 divided by 1, or exactly 3, matching the opposite measurement exactly, the signature of an isosceles right triangle. Carry the same setup to 90° instead with an opposite length of 10, and the hypotenuse settles at exactly 10 while the adjacent length collapses to 0, the shape flattened into a line.
Questions
What does 'opposite' mean in this calculator?
It's the length sitting directly across from the angle you entered — the one that never touches that angle's own vertex. In a shadow-free height reading, this is often the vertical measurement itself, such as a flagpole's height read against the sun's elevation.
How is this different from entering a side and the hypotenuse?
This sheet starts from an angle and a length measured directly across from it; a separate calculator on this site starts instead from any known length plus the hypotenuse together, then uses the Pythagorean theorem plus inverse sine to find everything else. Both approaches solve a full right triangle, just from a different pair of known measurements.
What happens as the angle gets close to 90°?
The adjacent length shrinks toward zero and the hypotenuse converges on the opposite measurement's own size. At exactly 90°, the shape has degenerated to a flat line — an opposite reading of 10 paired with that angle returns a hypotenuse of 10 and an adjacent value of 0.
Can the angle be 0° or 90° exactly?
0° breaks the calculation, since both sine and tangent vanish and dividing by zero sends the hypotenuse and the adjacent length toward infinity. 90° actually resolves cleanly — sine reaches its top value of 1 while tangent grows without bound, so the hypotenuse locks onto the opposite reading's own value and the adjacent measurement collapses to 0.
Why divide by sine to get the hypotenuse?
Because sine is defined as the opposite length divided by the hypotenuse, so rearranging that definition for the hypotenuse gives opposite divided by sine — the same relationship behind SOH in SOH-CAH-TOA, solved for the denominator instead of the ratio itself.
Does this work if my angle is given in radians?
Yes — switch the unit selector to radians or turns and enter the value there instead; the underlying formulas, hyp equals opposite divided by sine and adjacent equals opposite divided by tangent, stay identical no matter which unit is used to express the angle.