# Math: How to Prove the Pythagorean Theorem

*I hold both a bachelor's and a master's degree in applied mathematics.*

The Pythagorean theorem is one of the most well-known theorems in math. It is named after the Greek philosopher and mathematician Pythagoras, who lived around 500 years before Christ. However, most probably he is not the one who actually discovered this relationship.

There are signs that already 2,000 B.C. the theorem was known in Babylonia. Also, there are references that show the use of the Pythagorean theorem in India around 800 B.C. In fact, it is not even clear whether Pythagoras had actually anything to do with the theorem, but because he had a big reputation the theorem was named after him.

The theorem as we know it now was first stated by Euclid in his book *Elements* as proposition 47. He also gave a proof, which was quite complicated. It definitely can be proven a lot easier.

## What Is the Pythagorean Theorem?

The Pythagorean theorem describes the relationship between the three sides of a right triangle. A right triangle is a triangle in which one of the angles is exactly 90°. Such an angle is called a right angle.

There are two sides of the triangle that form this angle. The third side is called the hypothenuse. The Pythagorean states that the square of the length of the hypothenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides, or more formally:

Let a and b be the lengths of the two sides of a right triangle that form the right angle, and let c be the length of the hypothenuse, then:

**a ^{2} + b^{2} = c^{2}**

## The Proof of the Pythagorean Theorem

There are lots of proofs of the Pythagorean theorem. Some mathematicians made it a kind of sport to keep trying to find new ways to prove the Pythagorean theorem. Already, more than 350 different proofs are known.

One of the proofs is the rearranging square proof. It uses the picture above. Here we divide a square of length (a+b)x(a+b) into multiple areas. In both pictures, we see that there are four triangles with sides a and b forming a right angle and hypothenuse c.

On the left-hand side, we see that the remaining area of the square consists of two squares. One has sides of length a, and the other has sides of length b, which means that their total area is a^{2}+b^{2}.

In the picture on the righthand side, we see that the same four triangles appear. However, this time they are placed in such a way that the remaining area is formed by one square, which has sides of length c. This means that the area of this square is c^{2}.

Since in both pictures we filled the same area, and the sizes of the four triangles are equal, we must have that the sizes of the squares in the left picture add up to the same number as the size of the square in the above picture. This means that a^{2}+b^{2 }= c^{2}, and hence the Pythagorean theorem holds.

Other ways to prove the Pythagorean theorem include a proof by Euclid, using congruence of triangles. Furthermore, there are algebraic proofs, other rearrangement proofs and even proofs that make use of differentials.

## Pythagorean Triples

If a, b and c form a solution to the equations a^{2}+b^{2 }= c^{2} and a, b and c are all natural numbers, then a, b and c are called a Pythagorean triple. This means that it is possible to draw a right triangle such that all sides have an integer length. The most famous Pythagorean triple is 3, 4, 5, since 3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}. Other Pythagorean triples are 5, 12, 13 and 7, 24, 25. There are a total of 16 Pythagorean triples for which all numbers are less than 100. In total, there are infinitely many Pythagorean triples.

A Pythagorean triple can be created. Let p and q be natural numbers such that p < q. Then a Pythagorean triple is formed by:

a = p^{2} - q^{2}

b = 2pq

c = p^{2} + q^{2}

Proof:

(p^{2} - q^{2})^{2 }+ (2pq)^{2} = p^{4} - 2p^{2}q^{2} + q^{4} + 4p^{2}q^{2} = p^{4} + 2p^{2}q^{2} + q^{4} = (p^{2} + q^{2})^{2}

Furthermore, since p and q are natural numbers and p>q, we know that a, b and c are all natural numbers.

## Goniometric Functions

The Pythagorean theorem also provides the goniometric theorem. Let the hypothenuse of a right triangle have length 1 and one of the other angles be x then:

sin^{2}(x) + cos^{2}(x) = 1

This can be calculated using the formulas for the sine and cosine. The length of the adjacent side to the angle x is equal to the cosine of x divided by the length of the hypothenuse, which is equal to 1 in this case. Equivalently, the length of the opposite side has length cosine of x divided by 1.

If you want to know more about these kinds of calculations of angles in a right triangle, I recommend reading my article about finding the angle in a right triangle.

## Overview

The Pythagorean theorem is a very old mathematical theorem that describes the relationship between the three sides of a right triangle. A right triangle is a triangle in which one angle is exactly 90°. It states that a^{2} + b^{2} = c^{2}. Although the theorem is named after Pythagoras, it was known already for centuries when Pythagoras lived. There are a lot of different proofs for the theorem. The easiest uses two ways to divide the area of a square into multiple pieces.

When a, b and c are all natural numbers, we call it a Pythagorean triple. There are infinitely many of these.

The Pythagorean theorem has a close relation with the goniometric functions sine, cosine and tangent.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*