Pythagorean Theorem

The Pythagorean theorem is a fundamental geometric principle stating that in a right triangle, the square of the length of the longest side—called the hypotenuse (the side opposite the right angle)—equals the sum of the squares of the other two sides. If a and b are the two sides that form the right angle and c is the hypotenuse, the relationship is: a² + b² = c².

The theorem ranks among the most proven results in the history of mathematics. It underlies the measurement of distance in Euclidean geometry and has given rise to numerous principles used in statistics, engineering, and physics.

Mathematical form

When two sides are known, the third can be calculated. For example, if a = 3 and b = 4, then c² = 9 + 16 = 25, so c = 5. Sets of whole numbers satisfying this relationship—such as (3, 4, 5) or (5, 12, 13)—are known as Pythagorean triples.

The converse

The converse is also true: if a triangle's sides satisfy a² + b² = c², then the angle opposite side c is a right angle. This makes the theorem a method for verifying whether an angle is right, which is useful in construction and land measurement.

History

The name comes from Pythagoras, a Greek scholar who lived in the centuries before the Common Era, but the relationship predates him. Records from Babylon, Egypt, and India show that earlier mathematicians knew numerical sets satisfying the relationship, using them for purposes of measurement and construction. The formal geometric proof appears in Euclid's Elements.

Applications

The theorem extends to many domains. In coordinate mathematics, the distance between two points on a plane is derived from it: d = √((x₂−x₁)² + (y₂−y₁)²). There is also a general form that works for any triangle, called the law of cosines, of which the Pythagorean theorem is the special case when the angle is a right angle.