# Pythagoras' theorem

## In a nutshell

Pythagoras' theorem (or the *Pythagorean theorem*) is a formula that involves 3 sides of a right-angled triangle.

### Hypotenuse

The *hypotenuse* of a triangle is the longest side. For a right-angled triangle, the hypotenuse will be opposite the right angle.

## The formula for Pythagoras' theorem

Pythagoras' theorem states that

$a^2\,+b^2\,=c^2$

Where $c$ is the hypotenuse of the triangle, and $a$ and $b$ are the other 2 sides.

**Example 1**

*A right-angled triangle has the 2 shorter sides with length $5cm$ and $12cm$. Find the length of the longest side.*

*Sketch the triangle:*

*Label the sides:*

*Use Pythagoras' theorem:*

$a^2\,+b^2\,=c^2$**

*Replace $a$ and $b$ with your values, and solve:*

$12^2+5^2=c^2$

$144+25=c^2$

$c^2=169$

$c=\sqrt{169}=13$

$\underline{c=13cm}$

**Note: **Don't forget to write the correct unit for your length!

##### Example 2

*A right-angled triangle has a hypotenuse of length $8cm$, and a base of length $2cm$*. *What is the area of the triangle to two decimal places?*

*First, work out the perpendicular height of the triangle using Pythagoras' theorem. Let $a=2$ and $c=8$:*

*$a^2+b^2=c^2$*

*$2^2+b^2=8^2$*

*$b^2=8^2-2^2=64-4=60$*

*$b=\sqrt{60}cm$*

**

*Then, work out the area of the triangle using the area formula:*

*$\text{Area}=\dfrac{1}{2}\times\text{base}\times\text{perpendicular\,height}$*

*$\text{Area}=\dfrac{1}{2}\times2\times\sqrt{60}$*

*$\text{Area}=7.74596669...$*

*The area is *$\underline{7.75 \ cm^2 \space (2\thinspace d.p)}$*.*