The short answer
The Pythagorean theorem says a² + b² = c² for any right triangle, where c is the hypotenuse (the longest side, opposite the right angle) and a, b are the two legs. Given any two sides, solve for the third: c = √(a²+b²) to find the hypotenuse, or a = √(c²−b²) to find a leg.
Key takeaways
- The theorem only applies to right triangles — triangles with exactly one 90° angle.
- The hypotenuse is always the longest side and always stands alone on one side of the equation, never combined with a leg under the same square root.
- Pythagorean triples (3-4-5, 5-12-13, 8-15-17) are whole-number solutions that show up constantly in geometry problems and real-world construction.
- Multiplying every side of a Pythagorean triple by the same number produces another valid triple — 6-8-10 and 9-12-15 are both scaled from 3-4-5.
Solving for each side
| Unknown | Formula |
|---|---|
| c (hypotenuse) | c = √(a² + b²) |
| a (leg) | a = √(c² − b²) |
| b (leg) | b = √(c² − a²) |
Solving for a leg always subtracts under the square root, never adds — a leg is shorter than the hypotenuse, so removing the other leg's contribution from c² is what recovers it.
Common Pythagorean triples
| Triple | 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 |
Worked example: the 3-4-5 construction rule
3² + 4² = 9 + 16 = 25
√25 = 5
Builders measure 3 feet along one wall and 4 feet along a perpendicular wall from the same corner, then check the diagonal between those two points. A diagonal of exactly 5 feet confirms a true right angle — no protractor or angle finder needed, just a tape measure and this one identity.
Common mistakes to avoid
- Applying a² + b² = c² to a triangle that isn't actually right-angled — the identity simply doesn't hold without a genuine 90° angle.
- Adding instead of subtracting when solving for a leg — a leg comes from c² − (other leg)², not c² + (other leg)².
- Mislabeling which side is the hypotenuse — it's always the side opposite the right angle and always the longest, regardless of how the triangle is drawn or rotated.
- Mixing units before squaring — every side must be in the same unit (all feet, or all meters) before the formula gives a meaningful result.
Related calculators
- Right Triangle Calculator — solve for angles and area in addition to the missing side.
- Distance Calculator — see the same theorem applied to coordinate points instead of triangle sides.
- Triangle Calculator — work with triangles that aren't right-angled.
- Root Calculator — double-check the square root step by hand.