The Pythagorean theorem, or the Pythagoras theorem was named after the Greek mathematician Pythagoras, although it appears that thee theorem was in use for centuries before he was born in around 570 BC, and may have been discovered independently in many places around the world in ancient history. The Pythagorean theorem describes the relationship between lengths of sides in right triangles (not isosceles triangles, like Homer Simpson mistakenly thought).

Figure 1: The Pythagorean theorem applies to right triangles, which have a right angle. The longest side of the triangle opposite the right angle is known as the hypotenuse. The area of a square drawn from this side is equal to the sum of the squares drawn from the other two sides. This property allows for the length of any side of a right angle to be determined if the lengths of the other two sides are known.

The Pythagorean theorem is that the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the other two sides (Figure 1). The hypotenuse is the side of the right triangle opposite to the right angle. This trigonometric law is the simplest trigonometric law to understand, and is classically expressed as in (Eq. 1), where a and b are the lengths of the shorter two sides of the right triangle, and c is the length of the hypotenuse.

Eq. 1: \(a^2 + b^2 = c^2\)

Using the Pythagorean theorem, you can find the length of a hypotenuse if you know the length of the other two sides (Eq. 2 and Eq. 3). You just have to square the lengths of both sides, add them together (Eq. 2), and find the square root of the answer (Eq. 3).

Eq. 2: \(c^2 = 3^2 + 4^2 = 9 + 16 = 25\)

Eq. 3: \(c = \sqrt{25} = 5\)

Similarly, if you know the length of the hypotenuse and the length of one other side, you can find the length of the missing side (Eq. 4 and Eq. 5). You do this by finding the square of the hypotenuse, subtracting the sequare of the known side (Eq. 4), then find the square root of this number to give the length of the missing side.

Eq. 4: \(b^2 = 5^2 - 3^2 = 25 - 9 = 16\)

Eq. 5: \(b = \sqrt{16} = 4\)

The Pythagorean theorem is one of the oldest and best understood trigonometric laws is a good first place to start with understanding triangles and their properties.