Pythagorean Theorem
Find the hypotenuse or a missing leg of a right triangle.
The hypotenuse is the longest side, opposite the right angle.
The Pythagorean theorem says that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². So legs of 3 and 4 give c = √(9 + 16) = √25 = 5. Rearranged, a missing leg is b = √(c² − a²).
What the Pythagorean theorem does
The Pythagorean theorem relates the three sides of a right triangle — a triangle with one 90° angle. The two sides that form the right angle are the legs (a and b); the side opposite the right angle is the hypotenuse (c), and it is always the longest side. Square each leg, add them, and the result equals the square of the hypotenuse. That single relationship lets you find any one side from the other two.
a and b are the legs; c is the hypotenuse, opposite the right angle
Worked example
Find the hypotenuse of a right triangle with legs 3 and 4:
- 1 Identify the sides. The legs are a = 3 and b = 4; the unknown is the hypotenuse c.
- 2 Square each leg. 3² = 9 and 4² = 16.
- 3 Add the squares. 9 + 16 = 25, so c² = 25.
- 4 Take the square root. c = √25 = 5. To find a leg instead, rearrange to b = √(c² − a²).
Common Pythagorean triples
Whole-number side lengths that satisfy a² + b² = c² exactly.
| Leg a | Leg b | Hypotenuse c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
Only right triangles, and the converse
It applies only to right triangles. The relationship a² + b² = c² holds exactly when the triangle has a 90° angle. For triangles without a right angle you need the law of cosines instead.
The hypotenuse is special. It is the side opposite the right angle and always the longest of the three. When solving for a leg, the hypotenuse you enter must be longer than the known leg — otherwise no such triangle exists.
The converse test. The statement runs both ways: if the three sides of a triangle satisfy a² + b² = c² (with c the longest side), then the triangle must have a right angle. That is how builders confirm a square corner with a 3-4-5 measurement.