Pythagorean Theorem, theorem in two-dimensional geometry that states that the square of the length of the longest side of a right triangle equals the sum of the squares of the lengths of the other two sides. A right triangle is a triangle that contains one right (90°) angle. The longest side of a right triangle, called the hypotenuse, is the side opposite the right angle. The Pythagorean theorem is important in mathematics, physics, and astronomy and has practical applications in surveying.

If the legs of a right triangle are designated as a and b and the hypotenuse as c, the Pythagorean theorem can be written as an equation: a² + b² = c². Pythagorean triples are sets of three numbers, such as 3, 4, 5 or 1, 2, Ä that satisfy this equation and hence are possible lengths of sides of a right triangle:

The converse of the Pythagorean theorem is also true. That is, a triangle in which the square of one side equals the sum of the squares of the other two sides is a right triangle.

Although the theorem is associated with the name of the Greek mathematician Pythagoras, who lived in the 6th century bc, there is evidence that it was known at least 1,000 years before him. Babylonian texts dating from as early as 1800 bc contain discussions of Pythagorean triples.

The Greek mathematician Euclid formally proved the Pythagorean theorem at the end of Book I of his Elements, a text that was compiled about 300 bc. This work was the standard textbook for the study of geometry for about 2,000 years.