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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although knowledge of the theorem almost certainly predates him. The theorem is known in mainland China as the "Gougu theorem" (勾股定理) for the (3, 4, 5) triangle.

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

This is usually summarized as:

The square on the hypotenuse is equal to the sum of the squares on the other two sides.

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation

${\displaystyle a^{2}+b^{2}=c^{2}\,}$

or, solved for c:

${\displaystyle {\sqrt {a^{2}+b^{2}}}=c.\,}$

This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem.

History

The history of the theorem can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, and proofs of the theorem.

Megalithic monuments from circa 2500 BC in Egypt, and in the British Isles, incorporate right triangles with integer sides. Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.

Written between 2000–1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution i

1. Heath, Vol I, p. 144.
2. "Megalithic Monuments".
3. van der Waerden, B.L., Geometry and Algebra in Ancient Civilizations, Springer, 1983.