Pythagorean Theorem Calculator
Find the hypotenuse or any missing side of a right triangle using a² + b² = c².
Common Pythagorean Triples
| a | b | c (hypotenuse) | 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 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 9 | 40 | 41 | 81 + 1600 = 1681 |
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides (legs): a² + b² = c². It was known to Babylonian mathematicians 1,000 years before Pythagoras, but is named after the Greek mathematician who provided the first known formal proof.
What is a hypotenuse?
The hypotenuse is the longest side of a right triangle, always opposite the 90° right angle. In the Pythagorean theorem, c represents the hypotenuse. To find it, take the square root of the sum of the squared legs: c = √(a² + b²).
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². The most famous example is 3-4-5 (3² + 4² = 5² → 9 + 16 = 25). Other common triples include 5-12-13, 8-15-17, and 7-24-25. Any multiple of a Pythagorean triple is also a triple — e.g., 6-8-10 is double 3-4-5.
Can the Pythagorean theorem be used for non-right triangles?
No — the standard Pythagorean theorem only applies to right triangles. For other triangles, use the Law of Cosines: c² = a² + b² − 2ab × cos(C), where C is the angle opposite side c. The Law of Cosines reduces to the Pythagorean theorem when C = 90° (since cos(90°) = 0).
How do I find a missing leg when I know the hypotenuse and one leg?
Rearrange the formula: a = √(c² − b²). For example, if the hypotenuse is 13 and one leg is 5: a = √(169 − 25) = √144 = 12. Note that the hypotenuse must be longer than either leg — if it isn't, the inputs don't describe a valid right triangle.