Pythagorean theorem

In Euclidean geometry, the square of a right triangle’s hypotenuse equals the sum of the squares of its other two sides. It connects side lengths through the equation a² + b² = c².

Euclidean distance: The straight-line distance between two points in Euclidean space, computed from differences in their coordinates. The theorem derives this distance formula by applying the right-triangle relation to coordinate differences.
Right triangle: A triangle with one interior angle equal to 90 degrees. The theorem applies specifically to this triangle type, with its longest side opposite the right angle.
Pythagorean triple: Three positive integers satisfying a² + b² = c², such as 3, 4, and 5. Integer triples provide exact right-triangle dimensions useful in measurement and geometric constructions.
Babylonian mathematics: The mathematical practices of ancient Mesopotamian societies, preserved partly on clay tablets. Old Babylonian tablets record numerical knowledge of right triangles centuries before Pythagoras.
Non-Euclidean geometry: Geometries in which Euclid’s parallel postulate does not hold. On curved spaces, the Euclidean side-length equation generally fails for geodesic triangles.
Geometric proof of the Pythagorean theorem: A demonstration of the Pythagorean theorem using figures, areas, or geometric transformations. Area rearrangements make the equality visible without relying on algebraic manipulation.
Hypotenuse: The side opposite the right angle in a right triangle; it is the triangle’s longest side. The theorem identifies the square on this side with the combined squares on the legs.
Coordinate geometry: The study of geometric objects using coordinates and algebraic equations. The theorem supplies the standard formula for distances between points in a plane.
Plimpton 322: An Old Babylonian clay tablet listing numerical entries associated with right-triangle side ratios. Its entries demonstrate sophisticated triangle calculations long before the theorem's Greek attribution.
Spherical geometry: The geometry of points and great-circle arcs on a sphere. Spherical right triangles obey a trigonometric relation rather than the Euclidean sum of squared sides.

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