Pythagorean Theorem
The central relation of Euclidean geometry: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides:
a2 + b2 = c^2
where c is the hypotenuse. The theorem is named after the Greek mathematician Pythagoras, but it was known much earlier: Babylonian tablets such as Plimpton 322 (c. 1800 BCE) contain lists of integer triples satisfying the relation, and the earliest recorded proofs come from ancient Chinese and Indian traditions as well as from Euclid's Elements (c. 300 BCE), where it appears as Proposition I.47.
The statement has a geometric interpretation: the square constructed on the hypotenuse has area equal to the combined areas of the squares on the other two sides. One of the simplest proofs is by rearrangement — four copies of the triangle can be arranged inside a square in two ways, and comparing the leftover areas forces the identity. Hundreds of independent proofs are known, including proofs by dissection, algebra, and similarity.
The converse also holds: if the three side lengths of a triangle satisfy a2 + b2 = c^2, the angle opposite c is a right angle. This gives a practical tool for checking right angles — the familiar 3–4–5 triangle — used by builders and surveyors for millennia.
The theorem is the foundation of the Euclidean distance formula: the distance between two points (x1, y1) and (x2, y2) in a plane is sqrt((x2 − x1)2 + (y2 − y1)2). It underlies trigonometry (the identity sin2 θ + cos2 θ = 1 is the theorem applied to the unit circle), vector norms, and the geometry of coordinate systems. In non-Euclidean geometries the relation no longer holds in the same form, which is one way of seeing that the theorem encodes the flatness of Euclidean space.
Tags
geometry mathematics triangles