The Pythagorean theorem can be written as a single equation, but from it several practical forms follow, each handy in different situations. In this article we'll go through all of them.
In a right triangle, let us label:
Then the following holds:
In words: the sum of the squares of the legs equals the square of the hypotenuse.
If you know both legs and are looking for the hypotenuse, you solve for by taking the square root of both sides:
Example: ,📌 Remember: When computing the hypotenuse, and are always added.
If you know the hypotenuse and one of the legs, you compute the other by subtracting:
Example: ,📌 Remember: When computing a leg, you always subtract the square of the known leg from .
⚠️ The expression under the square root must be positive. If you get a negative number, you've mixed something up – most likely the hypotenuse and a leg.
The square in the formula is no accident – it comes from the area of a square. Originally, the Pythagorean theorem was proved geometrically: on each side of a right triangle you build a square, and the area of the square on the hypotenuse is exactly equal to the sum of the areas of the two squares on the legs.
👉 The classic geometric proof: Proof of the Pythagorean theorem and history
| We want | Formula |
| Hypotenuse | |
| Leg | |
| Leg | |
| Checking for a right triangle |