# Why Is the Pythagorean Theorem So Useful?

Updated: Aug 31

Author: Kelly Zeng

Editor: Kira Tian and He-Hanson Xuan

Artist: Susan Wu

The Pythagorean Theorem, a groundbreaking mathematical discovery, was named after Pythagoras, one of the most famous and controversial ancient Greek philosophers. The theorem states that the area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides. Thus, the formula a^2 + b^2 = c^2 was derived in which a and b are the lengths of the two legs, and c is the length of the hypotenuse.

The Pythagorean theorem is one of the most fundamental formulas in the field of mathematics. In school, students use it to find the length of an unknown side of a right triangle or the diagonal of a rectangle. In real life, it’s useful for two-dimensional navigation, such as air navigation because it helps pilots finding the shortest distance between two points. Architects, engineers, and security systems for facial recognition also use the Pythagorean Theorem.

After its advent, many mathematicians sought to prove this theorem in many ways–there are an estimated 370 proofs of this theorem. Rearrangement proof is probably the most well-known: two large squares each contain four identical triangles. The only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space must have equal area because their triangles are congruent. After only moving the position of the triangles, the area of the square with side c is equal to the sum of the area of squares a and b, which validates the formula.

A Pythagorean triple has three positive integers: a, b, and c, representing the legs and the hypotenuse. Composed in the form (a, b, c), some common triples are (3, 4, 5), (5, 12, 13), and (9, 12, 15). From this simple theorem, mathematicians derived many other math concepts, such as incommensurable lengths, Euclidean distance, and complex numbers. The Pythagorean Theorem remains one of the most prominent theorems in mathematics due to its straightforwardness and simplicity.

**Citations:**

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