How this instrument works
A 30-60-90 triangle is the shape you get by cutting an equilateral triangle exactly in half. Drop an altitude from one vertex of an equilateral triangle with side length 2, and it splits the base into two equal pieces of length 1 while dividing the apex angle into two 30° angles. Each half is a right triangle with hypotenuse 2 (the original side), a short leg of 1 (half the base), and — by the Pythagorean theorem, 2² − 1² = 3 — a long leg of √3. That single construction is the whole derivation, and it is why the ratio 1 : √3 : 2 never changes: every 30-60-90 triangle, at any size, is a scaled copy of that one bisected equilateral triangle.
Because the angles are locked at 30°, 60°, and 90°, one side length pins down the other two directly — there is no need to run the general Pythagorean theorem a² + b² = c² and solve for an unknown each time. Multiply the short leg (the side opposite the 30° angle) by 2 for the hypotenuse, and by √3 for the long leg opposite 60°. That shortcut is the entire reason this shape gets a name of its own: the fixed angle set, not just the side lengths, is doing work that a and b normally have to do case by case in an arbitrary right triangle.
One limit is worth noticing: as the short leg shrinks toward zero, the whole triangle collapses to a point, and every ratio still holds exactly, since 2 × 0 and 0 × √3 both vanish together. At the other end there is no ceiling; scale the short leg up to the radius of a hexagonal tile or the rise of a roof truss and the same two multipliers apply unchanged. That is the practical reason drafting set squares and roof-pitch tables are built around this exact shape rather than an arbitrary right triangle solved fresh each time.
- Enter the length of the known side into the Short leg (opposite 30°) field — this is the shortest of the three sides, the one across from the smallest angle.
- Read Hypotenuse for the longest side of the triangle, which is always exactly twice the short leg.
- Read Long leg (opposite 60°) for the middle-length side, the short leg multiplied by √3, roughly 1.732051.
- If you only know the hypotenuse or the long leg, divide backwards first — hypotenuse ÷ 2, or long leg ÷ √3 — to recover the short leg, then enter that figure.
Worked example — a short leg of 5
Take a 30-60-90 triangle with a short leg of 5 units, the side directly across from the 30° angle. The hypotenuse comes straight from the doubling rule: hyp = 2 × 5 = 10. There is no need to invoke the general Pythagorean theorem here; the fixed angle set already guarantees the 1 : √3 : 2 proportion, so doubling is the entire calculation for that side.
The long leg, opposite the 60° angle, follows the second rule: longLeg = 5 × √3 = 8.660254037844386. A quick check confirms the triangle is consistent: 5² + 8.660254037844386² = 25 + 75 = 100, which is 10² exactly — the Pythagorean theorem holds as a consequence of the fixed ratio, not as a separate computation you had to run yourself.
Questions
Where does the 1 : √3 : 2 ratio for a 30-60-90 triangle come from?
It comes from bisecting an equilateral triangle. Split an equilateral triangle with side 2 along its altitude and each half is a 30-60-90 triangle with hypotenuse 2, short leg 1, and — by the Pythagorean theorem, √(2² − 1²) — a long leg of √3. Every 30-60-90 triangle is a scaled copy of that one construction, which is why the ratio never changes.
How is a 30-60-90 triangle different from a 45-45-90 triangle?
A 45-45-90 triangle is isosceles: its two legs are equal, giving a ratio of 1 : 1 : √2. A 30-60-90 triangle has three different side lengths in ratio 1 : √3 : 2, because its two acute angles are unequal. Mixing up the two ratios — using √2 where √3 belongs, or doubling a leg that should stay equal — is the most common error made with either shape.
Which side is the short leg and which is the long leg?
The short leg sits opposite the 30° angle and is the shortest side; the long leg sits opposite the 60° angle and runs about 1.73 times the short leg. A useful check: in any triangle, a longer side always sits opposite a larger angle, so the hypotenuse, opposite the 90° angle, is necessarily the longest side of the three.
Do I still need the Pythagorean theorem for a 30-60-90 triangle?
Not for the two formulas here, since the fixed angles already fix the ratio. But the Pythagorean theorem is exactly what proves that ratio correct in the first place — shortLeg² + longLeg² always equals hyp² — and that identity is a fast way to sanity-check any answer this sheet returns.
Where do 30-60-90 triangles actually show up outside a textbook?
They appear anywhere a 60° angle meets a right angle: the altitude of an equilateral triangle or a hexagonal tile, a 30° roof pitch measured against vertical rise and horizontal run, and the standard 30-60-90 set square used in technical drafting. Each case reduces to the same short-leg-to-hypotenuse doubling used on this page.
How precise is the √3 used in the long leg calculation?
This sheet carries √3 to full double precision, 1.7320508075688772…, so a short leg of 5 returns a long leg of 8.660254037844386 with no rounding error beyond ordinary floating-point limits. Round only at the last step, when the figure gets written down for cutting or building.