Pythagorean theorem

In a right triangle, the two sides that meet at the 90-degree angle are called the legs; the side opposite that angle is the hypotenuse. If the leg lengths are a and b and the hypotenuse has length c, the theorem states a^2 + b^2 = c^2. Geometrically, the area of the square built on the hypotenuse equals the sum of the areas of squares built on the two legs. The relation applies to lengths in a Euclidean plane and depends on the triangle being right-angled.

The theorem can be rearranged to find a missing side: for example, c = sqrt(a^2 + b^2) when the hypotenuse is unknown. A triangle with legs 3 and 4 has hypotenuse 5 because 3^2 + 4^2 = 9 + 16 = 25 = 5^2. Whole-number side lengths that satisfy the relation, such as 3, 4, and 5, are called Pythagorean triples. The converse also holds: if the side lengths of a triangle satisfy a^2 + b^2 = c^2, where c is the longest side, then the angle opposite c is a right angle.

The theorem has many proofs, including demonstrations that rearrange areas and arguments based on similar triangles. Euclid's Elements gives a geometric proof in Book I, Proposition 47, by comparing the areas of figures constructed on the three sides. The theorem is one case of a broader relationship among the sides and angles of triangles: the law of cosines reduces to a^2 + b^2 = c^2 when the included angle is 90 degrees. In Cartesian coordinates, applying the theorem to horizontal and vertical differences gives the Euclidean distance formula between two points, d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

The familiar name links the theorem to Pythagoras, who lived in the sixth century BCE, but the mathematical relationship is older than the Greek tradition. Babylonian tablets from roughly 1900–1600 BCE show knowledge of numerical relationships involving right triangles. Related procedures and proofs also appear in ancient Indian and Chinese mathematical traditions. Euclid's Elements preserves a systematic Greek proof, but the historical evidence does not establish that Pythagoras himself discovered or first proved the theorem.

By connecting perpendicular side lengths to a diagonal, the theorem is useful in geometry, measurement, surveying, construction, and coordinate-based calculations. It is also a foundational result for later study of distance, trigonometry, and analytic geometry. Its many distinct proofs make it a longstanding example of how one mathematical statement can be established through different lines of reasoning.