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Pythagoras' Theorem

Learn Pythagoras' Theorem for GCSE Maths. Find the hypotenuse, shorter sides, and tackle 3D problems with step-by-step worked examples.

Tutor AI Team2026-02-24T03:53:42.207Z

Pythagoras' Theorem is one of the most famous results in mathematics. It describes the relationship between the three sides of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.

This theorem appears on both the Foundation and Higher tiers of the GCSE and is used in a huge range of problems — from calculating distances on coordinate grids to finding lengths in 3D shapes. Once you understand the core idea and can apply it confidently, you will find it crops up again and again.

In this guide, you will learn how to identify when to use Pythagoras' Theorem, how to find the hypotenuse, how to find a shorter side, and how to apply the theorem to real-world and three-dimensional problems.

Core Concepts

The Theorem

For any right-angled triangle with sides $a$ , $b$ and hypotenuse $c$ (the longest side, opposite the right angle):

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

This means if you know any two sides of a right-angled triangle, you can always find the third.

Identifying the Hypotenuse

The hypotenuse is always:

The longest side of the triangle
The side opposite the right angle

Before applying the theorem, always identify which side is the hypotenuse. This determines whether you add or subtract the squares.

Finding the Hypotenuse

When you need to find the longest side, rearrange the formula to:

$c = \sqrt{a^2 + b^2}$

Example: A right-angled triangle has shorter sides of 5 cm and 12 cm.

$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}$

Finding a Shorter Side

When you know the hypotenuse and one other side, rearrange to:

$a = \sqrt{c^2 - b^2}$

Example: A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm.

$a = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm}$

Pythagorean Triples

Some sets of whole numbers satisfy Pythagoras' Theorem exactly. The most common Pythagorean triples you should recognise are:

$3, 4, 5$ (and multiples: $6, 8, 10$ ; $9, 12, 15$ ; etc.)
$5, 12, 13$
$8, 15, 17$
$7, 24, 25$

Recognising these can save time in exams.

Pythagoras in 3D

To find the length of a space diagonal in a cuboid (or other 3D shape), apply Pythagoras' Theorem twice:

First, find the diagonal across the base using two dimensions.
Then use that diagonal with the height to find the space diagonal.

For a cuboid with dimensions $l$ , $w$ and $h$ , the space diagonal $d$ is:

$d = \sqrt{l^2 + w^2 + h^2}$

Pythagoras on a Coordinate Grid

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ :

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This is simply Pythagoras' Theorem applied to the horizontal and vertical distances.

Strategy Tips

Tip 1: Draw and Label the Triangle

Always sketch the right-angled triangle and label the known sides. Mark the right angle clearly. This helps you identify the hypotenuse and set up the calculation correctly.

Tip 2: Decide — Adding or Subtracting?

If you are finding the hypotenuse, you add the squares: $c^2 = a^2 + b^2$ .

If you are finding a shorter side, you subtract: $a^2 = c^2 - b^2$ .

Tip 3: Don't Forget the Square Root

A very common error is to calculate $a^2 + b^2$ and give that as the answer. Remember, you need to take the square root at the end to find the length.

Tip 4: Give Exact or Rounded Answers as Required

If the answer is not a whole number, leave it as a surd (e.g., $\sqrt{52}$ ) for an exact answer, or round to a specified number of decimal places or significant figures as the question requires.

Tip 5: Look for Hidden Right-Angled Triangles

In exam questions, the right-angled triangle is not always obvious. You might need to draw a perpendicular height in an isosceles triangle, or spot a right angle in a real-world diagram.

Worked Example: Example 1

Problem

A ladder of length 5 m leans against a vertical wall. The foot of the ladder is 3 m from the base of the wall. How high up the wall does the ladder reach?

Solution

The ladder, wall and ground form a right-angled triangle. The ladder is the hypotenuse ( $c = 5$ m), the distance from the wall is one shorter side ( $b = 3$ m), and the height $a$ is what we need to find.

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

$a^2 + 3^2 = 5^2$

$a^2 + 9 = 25$

$a^2 = 16$

$a = \sqrt{16} = 4 \text{ m}$

The ladder reaches 4 m up the wall.

Worked Example: Example 2

Problem

Find the length of the diagonal of a rectangle with length 8 cm and width 6 cm.

Solution

The diagonal of a rectangle splits it into two right-angled triangles. The diagonal is the hypotenuse.

$d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm}$

The diagonal is 10 cm.

Worked Example: Example 3

Problem

A cuboid has dimensions 3 cm by 4 cm by 12 cm. Find the length of the space diagonal. Give your answer as an exact value.

Solution

Method: Apply Pythagoras twice, or use the 3D formula directly.

Step 1: Find the diagonal of the base.

$d_1 = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm}$

Step 2: Use $d_1$ and the height to find the space diagonal.

$d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}$

The space diagonal is 13 cm.

Worked Example: Example 4

Problem

Find the distance between the points $A(1, 3)$ and $B(7, 11)$ .

Solution

$d = \sqrt{(7-1)^2 + (11-3)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ units}$

The distance is 10 units.

Worked Example: Example 5

Problem

An isosceles triangle has two equal sides of length 13 cm and a base of 10 cm. Find the perpendicular height of the triangle.

Solution

The perpendicular height bisects the base, creating two right-angled triangles. Each has hypotenuse 13 cm and base $\frac{10}{2} = 5$ cm.

$h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}$

The perpendicular height is 12 cm.

Practice Problems

Problem 1

A right-angled triangle has shorter sides of 7 cm and 24 cm. Find the hypotenuse.

Problem 2

A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the other shorter side.

Problem 3

A ship sails 12 km due north and then 9 km due east. How far is the ship from its starting point?

Problem 4

Find the distance between the points $P(-2, 1)$ and $Q(4, 9)$ .

Problem 5

A cuboid has dimensions 6 cm × 8 cm × 10 cm. Calculate the length of the space diagonal, correct to 1 decimal place.

Problem 6

An equilateral triangle has side length 8 cm. By splitting it in half, find its perpendicular height correct to 1 decimal place.

Want to check your answers and get step-by-step solutions?

Common Mistakes

Adding instead of subtracting when finding a shorter side. If the hypotenuse is known, you must subtract: $a^2 = c^2 - b^2$ .
Forgetting the square root. Calculating $a^2 + b^2$ and stopping there gives $c^2$ , not $c$ .
Misidentifying the hypotenuse. The hypotenuse is always opposite the right angle and is always the longest side.
Using the theorem on non-right-angled triangles. Pythagoras' Theorem only works for right-angled triangles. For other triangles, you need the cosine rule.
Rounding too early. Keep full calculator values through intermediate steps and only round at the very end.

Frequently Asked Questions

When should I use Pythagoras' Theorem?

Use it whenever you have a right-angled triangle and know two of the three sides, and need to find the third. Also use it for distance problems on coordinate grids.

How do I know if a triangle is right-angled?

If you are given three side lengths, check whether the square of the longest side equals the sum of the squares of the other two. If $a^2 + b^2 = c^2$ , it is right-angled.

What is a surd answer?

A surd is an exact answer left in square root form, such as $\sqrt{52}$ or $2\sqrt{13}$ . Questions asking for an exact answer expect you to simplify the surd where possible rather than giving a decimal.

Can Pythagoras' Theorem give a negative answer?

No. Since you are finding a length, the answer is always positive. If you get a negative number under the square root, you have made an error — most likely subtracting the wrong way round.

Do I need to memorise $a^2 + b^2 = c^2$?

Yes. This formula is not given on the GCSE formula sheet. You must know it from memory.

Key Takeaways

✓
The core formula. For a right-angled triangle: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
✓
Hypotenuse = add and root. To find the hypotenuse, add the squares and take the square root.
✓
Shorter side = subtract and root. To find a shorter side, subtract the squares and take the square root.
✓
Works in 3D too. Apply Pythagoras twice to find space diagonals in cuboids and other 3D shapes.
✓
Distance formula. The distance between two coordinate points is just Pythagoras applied to horizontal and vertical differences.
✓
Always draw a diagram. Sketching the triangle helps you identify sides correctly and avoid errors.

Tags:

#gcse#maths#geometry#pythagoras

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Core Concepts

The Theorem

Identifying the Hypotenuse

Finding the Hypotenuse

Finding a Shorter Side

Pythagorean Triples

Pythagoras in 3D

Pythagoras on a Coordinate Grid

Strategy Tips

Tip 1: Draw and Label the Triangle

Tip 2: Decide — Adding or Subtracting?

Tip 3: Don't Forget the Square Root

Tip 4: Give Exact or Rounded Answers as Required

Tip 5: Look for Hidden Right-Angled Triangles

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Probability Basics

Expanding and Factorising

Fractions, Decimals and Percentages

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