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