Pythagorean Theorem

The Pythagorean theorem describes a fundamental relationship between the three sides of a right-angled triangle, that is, a triangle containing one 90-degree angle. It states that the area of the square built on the hypotenuse, the longest side lying opposite the right angle, equals the combined area of the squares built on the two shorter sides, called the legs. In other words, once you know any two of the sides, you can always work out the third. The relationship holds only for right triangles, and its converse is equally useful, since a triangle whose sides satisfy it must contain a right angle. Named after the ancient Greek mathematician Pythagoras, it is one of the oldest and most thoroughly proven results in all of mathematics, with hundreds of distinct proofs. It underpins the way we measure straight-line distance between two points and is essential to trigonometry, coordinate geometry, and vectors. In everyday practice it is used by builders, carpenters, and surveyors to check that corners are truly square and to lay out level foundations. It also appears throughout navigation, engineering, computer graphics, and physics, making it one of the most widely applied ideas ever discovered.