Over 2,000 years ago, a Greek philosopher called Pythagoras created a very famous theorem about triangles. It **lets you work out the length of any side in a right-angled triangle**!

**In this blog, jump to:**

- What is the Pythagorean theorem?
- Pythagorean theorem formula
- How to use the Pythagorean theorem
- — How to find the hypotenuse of a right-angle triangle
- — How to find the shortest side of a right-angle triangle
- Pythagorean theorem examples

## What is the Pythagorean theorem?

**The Pythagorean theorem says:**

*“In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.”*

This may sound a bit complicated, but it makes a lot more sense when we look at what it’s saying in the form of a picture:

When squares are drawn along each side in a right-angled triangle, the **total area of the two smaller squares** (those along the shortest sides, **a** and **b**) **is the same as the area of the largest square** (the square along the longest side, **c**).

By working out the area of each square, we can find the length of any side in a right-angled triangle. And luckily, Pythagoras created a handy formula to help us do this!

## Pythagorean theorem formula

The formula for Pythagoras’ theorem is **a² + b² = c²**.

In this equation, **c** is the longest side of a right-angled triangle. This line is also known as the hypotenuse.

**A** and **b** represent the other two sides of the triangle.

To use the formula, **we need to know the length of any two sides in a right-angled triangle**. We can then rearrange the formula to find the side we’re looking for.

So, if we take the formula **a² + b² = c²**…

We can rearrange it to help us find the length we’re missing:

To find the length of a: **a² = c² – b²**

To find the length of b: **b² = c² – a²**

To find the length of c: **c² = a² + b²**

## How to rearrange the Pythagorean theorem

- We start off by
**squaring the two lengths we do know**to work out the area of their corresponding squares. - We then
**add or minus the area of these squares**(depending on the formula used). - Finally, we
**square root the result to find the length of the missing side**.

This may sound complicated, but don’t worry! It’s easy to use once you know how. Let’s take a look at some examples.

## How to use the Pythagorean theorem

**How to find the hypotenuse of a triangle**

If we don’t know the length of the hypotenuse (longest side) of a right-angled triangle (**c**), we can work it out using Pythagoras’ theorem.

To do this, we first need to rearrange the equation to find c:

**a² + b² = c²**

becomes **c² = a² + b²**

Now, let’s take a look at the following example:

In this example, we know that side **a **is 4cm and side **b **is 3cm.

**1. Using our rearranged equation, we first need to work out a² (4²) + b² (3²):**

c² = a² + b²

c² = 4² + 3²

c² = 16 + 9

**2. Then, we can add these two values together. This gives us…**

c² = 25

**3. Now we’re in the final stretch. We just need to find the square root of 25 to find the length of c!**

c² = √25

**c = 5cm**Try some questions in DoodleMaths

**How to find a shorter side in a right-angled triangle**

We can also use Pythagoras’ theorem when we don’t know the length of one of the shorter sides of a right-angled triangle.

We can rearrange the formula to help us find the side we don’t know:

**a² + b² = c²**

**a² = c² – b²** (if we don’t know a) OR **b² = c² – a² **(if we don’t know b)

**1. In this example, we’re trying to work out the length of side a. So, we can rearrange the formula to become:**

a² = c² – b²

In this example, we know that c is 5cm long and b is 3cm long.

a² = 5² – 3²

**2. Next, we need to square 5 (c) and 3 (b) to work out the area of the squares along their sides.**

a² = 5² – 3²

a² = 25 – 9

**3. Once we’ve done this, we need to minus 25 from 9 (b from c):**

a² = 16

**4. Now we’re nearly there! If we find the square root of 16, we’ll have our answer.**

a = √16

**a = 4cm**

Try some questions in DoodleMaths

## Pythagorean theorem examples

Ready to look at some more examples or have a go at using the theorem yourself? Take a look at these example questions!

**Example 1: find the length of b**

Why not have a go at the above example yourself? When you’re ready, take a look below for the answer!

b² = c² – a²

b² = 10² – 8²

b² = 100 – 64

b² = 36

b = √ 36

b = **6cm**

**Example 2: find the length of c**

c² = a² + b²

c² = 8² + 6²

c² = 64 – 36

c² = 100

c = √ 100

c = **10cm**

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