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What is the Pythagorean theorem? Examples and how to use it

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!

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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

  1. We start off by squaring the two lengths we do know to work out the area of their corresponding squares.
  2. We then add or minus the area of these squares (depending on the formula used).
  3. 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.

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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 = 5cmTry 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

Master the Pythagorean theorem with DoodleMaths

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Powered by Proxima, our adaptive algorithm, it creates every child a personalised work programme tailored to their strengths and weaknesses.

Designed to be used for just 10 minutes a day, it revises tricky topics and consolidates what they already know, boosting their confidence a little every day.

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