Pythagoras Theorem

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Pythagoras Theoremvideo

Pythagoras Theorem

Pythagoras theorem

Who is Pythagoras?

Pythagoras was a Greek mathematician who lived over 2500 years ago
He is most famous for Pythagoras’ theorem, which includes the important formula for right-angled triangles

What is Pythagoras' theorem?

Pythagoras' theorem is a formula that links the lengths of the three sides of a right-angled triangle
The longest side of a right-angled triangle is called the hypotenuse
- The hypotenuse will always be the side opposite the right angle
Pythagoras' theorem states that $a^{2} + b^{2} = c^{2}$
- $c$ is the length of the hypotenuse
- $a$ and $b$ are the lengths of the two shorter sides
  - It does not matter which is labelled $a$ and which is labelled $b$

A right-angled triangle with the sides labelled a, b and c

How do I use Pythagoras’ theorem to find the length of the hypotenuse?

To find the length of the hypotenuse
- Square the lengths of the two shorter sides
- Add these two numbers together
- Take the positive square root
This can be written as $c = \sqrt{a^{2} + b^{2}}$
- This is just a rearrangement of the formula $a^{2} + b^{2} = c^{2}$ to make $c$ the subject
- Note that when finding the hypotenuse you add inside the square root

How do I use Pythagoras’ theorem to find the length of a shorter side?

To find the length of a shorter side
- Square the lengths of the hypotenuse and the other shorter side
- Subtract these numbers to find the difference
- Take the positive square root
This can be written as $a = \sqrt{c^{2} - b^{2}}$
- This is just a rearrangement of the formula $a^{2} + b^{2} = c^{2}$ to make $a$ the subject
- Note that when finding one of the shorter sides you subtract inside the square root

Can I use Pythagoras’ theorem with other shapes?

You can use Pythagoras' theorem with any shape that can be split into right-angled triangles
You can find the length of the diagonal of a rectangle or square
- Use its diagonal to split the shape into two identical right-angled triangles
- The diagonal is now the hypotenuse of the right-angled triangles

Worked example

In the following diagram:
$A B = 12 cm$
$A C$ is a straight line, with $A D = 9 cm$ and $A C = 22 cm$

Find $x$ , the length of side $B C .$ Give your answer to 1 decimal place.

Answer:

To find $x$ , we first need to find the length of $B D$ using triangle $A B D$
Note that $B D$ is a shorter side
Apply Pythagoras' theorem, $a = \sqrt{c^{2} - b^{2}}$

$B D = \sqrt{12^{2} - 9^{2}} = \sqrt{63} = 7.93725 . . .$

It is best to leave rounding until the very end, use $\sqrt{63}$ (or $3 \sqrt{7}$ if this is what your calculator has given you) in subsequent working

Find the length of $D C$ by subtracting the length of $A D$ from the length of $A C$

$D C = 22 - 9 = 13 cm$

Now we can find $x$ using triangle $B C D$
Note that $B C$ is the hypotenuse
Apply Pythagoras' theorem, $c = \sqrt{a^{2} + b^{2}}$

$x = \sqrt{B D^{2} + D C^{2}} = \sqrt{{(\sqrt{63})}^{2} + 13^{2}} = \sqrt{63 + 169}$

$x = \sqrt{232} = 15.23154621 . . .$

Round to 1 decimal place

$x = 15.2 cm$