Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse, the side opposite the right angle, equals the sum of the squares of the lengths of the other two sides, a relationship expressed algebraically as a squared plus b squared equals c squared. The theorem is traditionally attributed to the ancient Greek mathematician and philosopher Pythagoras, active around the sixth century BCE, and it is named for him, though surviving evidence shows that the same numerical relationship was already known and used, without a general proof, by earlier Babylonian and other ancient mathematicians well before his time. The oldest surviving complete mathematical proof of the theorem appears in Euclid's Elements, compiled around 300 BCE, which presents a rigorous geometric demonstration, and many hundreds of distinct alternative proofs of the theorem have since been published using a wide range of different mathematical approaches. The Pythagorean theorem is considered one of the most fundamental results in elementary geometry, underlying the standard mathematical definition of distance in Euclidean space, and it remains foundational to trigonometry, physics and countless practical applications in fields such as construction, navigation and surveying.