Pythagorean Theorem — a² + b² = c²
Variable Key
Find hypotenuse c
Given both legs a and b.
Find leg a or b
Given the hypotenuse and one leg.
Pythagorean triples
Integer right triangles (whole number sides).
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written as a² + b² = c², where c is the hypotenuse.
Tip: When two sides have a ratio close to 3:4, check if the third side is 5× the GCD. This instantly confirms a 3-4-5 family triple.
- 1Identify the right angle (90°) in the triangle
- 2Label the two shorter sides a and b, and the longest side (opposite 90°) as c
- 3Square each of a and b, then add them
- 4Take the square root of the sum to find c
Pythagorean triples
These are sets of three positive integers (a, b, c) that perfectly satisfy the theorem. The most famous is 3-4-5. Multiples also work: 6-8-10, 9-12-15, etc.
Checking if a triangle is right-angled
If a² + b² = c² exactly, the triangle is right-angled. If a² + b² < c², it is obtuse. If a² + b² > c², it is acute.
| a | b | c | Verify |
|---|---|---|---|
| 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 ✓ |
Fun Fact
There are over 370 known proofs of the Pythagorean theorem — more than any other theorem in mathematics. One was written by US President James Garfield in 1876.