The theorem
a² + b² = c²
c = √(a² + b²) a = √(c² − b²) b = √(c² − a²)
In any right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides. With legs of 3 and 4: 3² + 4² = 9 + 16 = 25, and √25 = 5. It only works for right triangles; for any other triangle use the law of cosines, c² = a² + b² − 2ab·cos C.
Pythagorean triples
Whole-number sides that satisfy the theorem are called Pythagorean triples. Builders have used 3-4-5 for thousands of years to lay out square corners: measure 3 units along one wall, 4 along the other, and adjust until the diagonal is exactly 5.
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| 9 | 40 | 41 |
| 12 | 35 | 37 |
| 11 | 60 | 61 |
Everyday uses
A TV's size is its diagonal: a 16:9 screen 48 inches wide and 27 inches tall measures √(48² + 27²) ≈ 55 inches. A ladder 13 feet long placed 5 feet from a wall reaches √(13² − 5²) = 12 feet up. The straight-line distance on a map grid, and the distance formula in coordinate geometry, are the same theorem.
History
The relationship was known long before Pythagoras (about 570 to 495 BC): the Babylonian tablet Plimpton 322, from around 1800 BC, lists fifteen triples, and it appears in ancient Indian and Chinese texts. The theorem carries Pythagoras's name because his school is traditionally credited with the first proof. Euclid's Elements gives the classic proof in Book I, Proposition 47.