The Theorem
a² + b² = c²
where c is the hypotenuse (longest side, opposite the 90° angle)
Attributed to Pythagoras (~570–495 BC), though Babylonian tablets show it was known at least 1,000 years earlier. It's one of the most proven theorems in mathematics—over 400 distinct proofs exist, including one by James Garfield (before he became US President).
Common Pythagorean Triples
| a | b | 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 ✓ |
| 20 | 21 | 29 | 400 + 441 = 841 ✓ |
Any multiple of a triple also works: 3-4-5 gives 6-8-10, 9-12-15, 30-40-50, etc.
Practical Applications
Construction: The 3-4-5 method checks if corners are square. Measure 3 feet along one wall, 4 feet along the other—if the diagonal is exactly 5 feet, the corner is 90°.
Navigation: Finding straight-line distance between two points that are offset horizontally and vertically. Walk 300m east and 400m north—you're 500m from the start in a straight line.
Screen sizes: A 16:9 display has sides in the ratio 16:9. For a 55" TV (diagonal), width = 55 × 16/√(16² + 9²) ≈ 47.9", height ≈ 27.0".