Special Right Triangles Calculator

Background.

Cut a square along its diagonal and you get two 45-45-90 triangles. Drop a line from one corner of an equilateral triangle to the middle of the opposite side and you get two 30-60-90 triangles. Those two shapes are the special right triangles, and they are special because their sides stand in fixed, exact ratios: once you know one side, the other two follow without any trigonometry.

For the half-square the ratio is 1 : 1 : √2. Both legs are the side of the square, and the hypotenuse is its diagonal, about 1.414 times longer. For the half-equilateral triangle it is 1 : √3 : 2. The short leg is half of the original side and the hypotenuse is the whole side, so the hypotenuse is always exactly twice the short leg, while the long leg, the triangle's altitude, is √3 or about 1.732 times the short leg.

These ratios are where the exact trigonometric values come from: sin 30° = 1/2, cos 45° = √2/2 and tan 60° = √3 can all be read straight off the sides. They are also built into everyday tools. A drafting kit has one set square of each shape, and a hexagonal nut is six equilateral triangles, so the distance across its flats is √3 times the length of one side.

Pick the triangle, enter the short leg, and the page multiplies through. In the 30-60-90 case the leg you enter must be the one opposite the 30° angle; entering the long leg by mistake makes every answer √3 times too large.

What is special right triangles calculator?

A special right triangle is a right triangle whose angles fix its sides in a ratio that can be written exactly with square roots. The 30-60-90 triangle, half of an equilateral triangle, has sides in the ratio 1 : √3 : 2. The 45-45-90 triangle, half of a square and the only isosceles right triangle, has sides in the ratio 1 : 1 : √2. Any one side therefore determines the other two.

How to use this calculator.

1. 01Choose 30°–60°–90° or 45°–45°–90° from the Special triangle list.
2. 02Enter the short or reference leg. For a 30-60-90 triangle this is the side opposite the 30° angle, the shortest side. For a 45-45-90 triangle either leg will do, since the two are equal.
3. 03If you only know the hypotenuse, divide it by 2 for a 30-60-90 triangle or by √2 for a 45-45-90 triangle, then enter that leg. If you know the long leg of a 30-60-90 triangle, divide it by √3.
4. 04Read the hypotenuse and the other leg. They come out in the same unit as the leg you entered, whether centimetres, inches or anything else.

The formula.

For a 30-60-90 triangle with short leg a: hypotenuse = 2a and long leg = a√3. For a 45-45-90 triangle with leg a: hypotenuse = a√2 and other leg = a. Both follow from the Pythagorean theorem. In the half-equilateral triangle the hypotenuse is the full side 2a, so the long leg is √((2a)² − a²) = √(3a²) = a√3. In the half-square both legs are a, so the hypotenuse is √(a² + a²) = a√2. Because every side is a fixed multiple of a, doubling the leg doubles every output; the only thing that changes the shape is the choice of triangle. √3 ≈ 1.7320508076 and √2 ≈ 1.4142135624 are irrational, so the decimals shown are rounded, while the radical forms 2√3 or 3√2 are exact.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.