# The Pythagorean Theorem

Introduction

The Pythagorean Theorem is probably the most famous theorem in Geometry. Even elementary school pupils know it by heart. It states that the given a right triangle with legs $a$, $b$, and hypotenuse $c$$a^2 + b^2 = c^2$

Although it is one of the easiest theorems to prove, its “cousin theorem”, the Fermat’s Last Theorem, is also one of the hardest mathematical problems of all time. It has puzzled mathematicians for more than 300 years, until Andrew Wiles proved it in 1995. Wiles spent about 8 years solving the problem.

In this post, we learn about one of the simplest proofs of the Pythagorean Theorem — and my favorite too.

Pythagorean Theorem

Given a right triangle with legs $a$ and $b$ and hypotenuse c, $a^2 + b^2 = c^2$

Proof

Create four right triangles of the same size with side lengths $a$, $b$, and $c$, where $c$ is the hypotenuse. Form a square with side length $a + b$ as shown in the first diagram in the figure below. Clearly, the area of the white square is $c^2$ (prove it!).

Now, maintaining the width of the first square and rearranging the triangles as shown in the second diagram, two white squares are formed: one with area $a^2$ and the other with area $b^2$. Since we just rearrange the triangles, the area not covered by the square in the first diagram is equal to the area not covered by the squares in the second diagram. Therefore, $a^2 + b^2 = c^2$.