Proving the Pythagorean Theorem

Some algebraic and geometric proofs of the best-known theorem of all time

A right triangle is a triangle in which one of the three angles is 90°. The triangle shown below is right-angled because the angle ⦣ACB between sides a and b is 90°. The side opposite the right angle, c, is called the hypotenuse.

One of the most important and best-known theorems of Euclidean geometry, the Pythagorean theorem, expresses a fundamental property of right-angled triangles. It states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. The equation c² = a²+b² expresses this relationship in mathematical symbols.

Throughout history, countless proofs of the Pythagorean theorem have been given. Below we will examine some of the best known.

The simplest and most convincing proof

The simplest and most immediately convincing proof is a geometric one based on rearranging the position of four triangles within a square, as shown in the following image.