Pick the triangle, say which side you know and enter its length. You'll get every side exactly, like 5√3, and as a decimal.
The 30-60-90 triangle formula
short leg : long leg : hypotenuse = x : x√3 : 2x
The short leg is opposite the 30° angle, the long leg is opposite 60°, and the hypotenuse is opposite the right angle. If the short leg is 5, the long leg is 5√3 ≈ 8.6603 and the hypotenuse is 10.
- Know the hypotenuse? Halve it for the short leg: hypotenuse 12 → short leg 6, long leg 6√3.
- Know the long leg? Divide by √3 for the short leg: long leg 9 → short leg 9/√3 = 3√3 ≈ 5.1962.
The 45-45-90 triangle formula
leg : leg : hypotenuse = x : x : x√2
Both legs are equal, and the hypotenuse is a leg times √2. A leg of 4 gives a hypotenuse of 4√2 ≈ 5.6569. Going the other way, divide by √2: a hypotenuse of 10 gives legs of 10/√2 = 5√2 ≈ 7.0711.
Why it works
A 30-60-90 triangle is half of an equilateral triangle, so the short leg is half the hypotenuse. A 45-45-90 triangle is half of a square, cut along the diagonal. In both, the Pythagorean theorem gives the third side: x² + (x√3)² = 4x² = (2x)².
How we checked this
Before every update, this calculator is run in a real browser with these known answers, and it must match all of them:
- 30-60-90, short leg 5: long leg 5√3, hypotenuse 10 ✓
- 30-60-90, hypotenuse 12: long leg 6√3 = 10.39230485 ✓
- 45-45-90, hypotenuse 10: legs 5√2 = 7.071067812 ✓
Last checked October 26, 2026. Spotted something wrong? The math is done in your browser with the formula shown on this page, so you can check any result by hand.
Questions people ask
What is the 30-60-90 triangle rule?
The sides are in the ratio 1 : √3 : 2. The hypotenuse is twice the short leg and the long leg is the short leg times √3.
What is the 45-45-90 triangle rule?
The legs are equal and the hypotenuse is a leg times √2, a ratio of 1 : 1 : √2.
Which leg is the short leg?
The side opposite the 30° angle. It's always the shortest side and half the hypotenuse.
How do I find the legs of a right triangle in general?
If it isn't a special triangle, use the Pythagorean theorem: a² + b² = c². The Pythagorean theorem calculator does this for any right triangle.