How to Find the Hypotenuse (Pythagorean Theorem)

The theorem

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared + b squared = c squared, where c is the hypotenuse. Rearranged, the hypotenuse is the square root of (a squared + b squared).

A worked example

With legs of 3 and 4: c = square root of (3 squared + 4 squared) = square root of (9 + 16) = square root of 25 = 5. That 3-4-5 triangle is the most famous right triangle because the sides come out as whole numbers. The Pythagorean Calculator finds the hypotenuse — or a missing leg — for any right triangle.

Finding a missing leg

The theorem works backwards too. If you know the hypotenuse and one leg, rearrange: leg = square root of (c squared - a squared). Know the hypotenuse is 13 and one leg is 5, and the other is the square root of (169 - 25) = the square root of 144 = 12 — another whole-number triple.

Where it shows up

Right-triangle math is everywhere practical: checking that a corner is truly square in carpentry (the 3-4-5 trick), finding the straight-line distance between two points, sizing a diagonal TV or screen, and working out a roof or ramp slope. Any time a right angle is involved, the hypotenuse is a square root away.

Frequently asked questions

Does it work for any triangle? No — only right triangles; other triangles need the law of cosines.

What is a Pythagorean triple? Three whole numbers that fit the theorem, like 3-4-5 or 5-12-13.

How do I use it to check a square corner? Measure 3 units along one edge and 4 along the other; if the diagonal is exactly 5, the corner is square.