Find any side of a right triangle using the Pythagorean theorem. Solve for the hypotenuse or either leg.
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is arguably the most widely recognized mathematical relationship in history, appearing in ancient Babylonian tablets from 1800 BCE, Egyptian construction techniques, and the work of Chinese mathematicians — all independently, centuries before Pythagoras formalized it in the 6th century BCE.
The theorem's power lies in its universality: it works for every right triangle, regardless of size. A right triangle with legs of 3 cm and 4 cm has a hypotenuse of 5 cm. Scale it up — legs of 300 km and 400 km — the hypotenuse is 500 km. The ratio is always 3:4:5. This scaling property makes it invaluable in architecture, navigation, engineering, and everyday construction.
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the theorem: a² + b² = c². The most famous is (3, 4, 5): 9 + 16 = 25. Others include (5, 12, 13): 25 + 144 = 169; (8, 15, 17): 64 + 225 = 289; and (7, 24, 25): 49 + 576 = 625. Any multiple of a triple is also a triple: (6, 8, 10), (9, 12, 15), and (12, 16, 20) are all derived from (3, 4, 5).
Ancient Egyptian builders reportedly used ropes tied in 12 equal segments to form a 3-4-5 triangle, guaranteeing perfect right angles for pyramid construction. This "rope stretching" technique required no knowledge of the theorem's formal proof — just the empirical observation that a 3-4-5 triangle is always right-angled.
The Pythagorean theorem has more known proofs than any other theorem in mathematics — over 370 distinct proofs have been documented. They range from geometric rearrangements to algebraic derivations to trigonometric proofs to calculus-based proofs. President James Garfield discovered an original proof in 1876, years before his presidency.
The visual proof by dissection is particularly intuitive: arrange four identical right triangles inside a large square. The area inside the large square that is not covered by triangles equals c². Rearrange the same four triangles within the same square differently, and the uncovered area now consists of two smaller squares with areas a² and b². Since the total area hasn't changed, a² + b² = c².
The distance formula in two dimensions is the Pythagorean theorem applied to coordinates: d = √((x₂-x₁)² + (y₂-y₁)²). In three dimensions, it extends naturally: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²). In n dimensions, you simply add n squared difference terms under the radical sign. This generalization is called Euclidean distance and forms the basis of distance-based machine learning algorithms.