Find the length of the hypotenuse.
Use 3² + 4² = x².
3² + 4² = 9 + 16 = 25, so x = 5 cm.
Pythagoras’ Theorem
Pythagoras’ theorem is used to find missing side lengths in right-angled triangles. It is essential in geometry and links directly to trigonometry and coordinates.
Overview
Pythagoras' Theorem is used to find a missing side in a right-angled triangle.
It only works when one angle is exactly \(90^\circ\).
\( a^2 + b^2 = c^2 \)
The side called \(c\) is always the hypotenuse, which is the longest side and is opposite the right angle.
What you should understand after this topic
- Understand when Pythagoras' Theorem can be used
- Identify the hypotenuse correctly
- Find a missing longer side
- Find a missing shorter side
- Check answers correctly
Key Definitions
Right-Angled Triangle
A triangle with one angle equal to \(90^\circ\).
Hypotenuse
The longest side, opposite the right angle.
Shorter Sides
The two sides that form the right angle.
Square Number
A number multiplied by itself, for example \(5^2 = 25\).
Square Root
The value that multiplies by itself to make the number.
Pythagorean Triple
A set of whole numbers that fit Pythagoras' theorem, such as \(3, 4, 5\).
Key Rules
Use only in right-angled triangles
If there is no \(90^\circ\) angle, you cannot use it.
Identify the hypotenuse first
This is always opposite the right angle.
Main formula
\( a^2 + b^2 = c^2 \)
Finding a shorter side
Rearrange to subtract: \( a^2 = c^2 - b^2 \)
Quick Formula Guide
Find the hypotenuse
\( c = \sqrt{a^2 + b^2} \)
Find a shorter side
\( a = \sqrt{c^2 - b^2} \)
Hypotenuse rule
It is always the longest side.
Calculator step
Do the square root at the end, not too early.
How to Solve
Step 1: Check for a right-angled triangle
Pythagoras' theorem only works in right-angled triangles.
Exam tip: Always check for the \(90^\circ\) angle first.
Step 2: Identify the hypotenuse
The hypotenuse is the longest side and is opposite the right angle.
\( a^2 + b^2 = c^2 \)
\(c\) is the hypotenuse.
\(a\) and \(b\) are the shorter sides.
Step 3: Find the hypotenuse
If the missing side is the hypotenuse, add the squares.
\( c = \sqrt{a^2 + b^2} \)
Exam thinking: Missing longest side → add.
Step 4: Find a shorter side
If the missing side is not the hypotenuse, subtract.
\( a = \sqrt{c^2 - b^2} \)
Exam thinking: Missing shorter side → subtract.
Step 5: Substitute carefully
Always write the formula first, then substitute.
\( 3^2 + 4^2 = c^2 \)
\( 9 + 16 = c^2 \)
\( c = 5 \)
Step 6: Recognise Pythagorean triples
\(3, 4, 5\)
Most common triple.
\(5, 12, 13\)
Common exam triple.
\(8, 15, 17\)
Useful to recognise.
\(7, 24, 25\)
Higher-level triple.
Step 7: Exam method summary
You will need powers and roots for square roots.
- Check the triangle is right-angled.
- Identify the hypotenuse.
- Decide: add or subtract.
- Square values.
- Calculate.
- Square root the answer.
Example Questions
Edexcel
Exam-style questions focusing on finding the hypotenuse in right-angled triangles.
Edexcel
A right-angled triangle has shorter sides 6 cm and 8 cm.
Find the hypotenuse.
Edexcel
A right-angled triangle has shorter sides 7 cm and 24 cm.
Find the hypotenuse.
AQA
Exam-style questions focusing on finding missing shorter sides and checking right-angled triangles.
AQA
The hypotenuse is 10 cm and one shorter side is 6 cm.
Find the other shorter side.
AQA
A triangle has sides 5 cm, 12 cm and 13 cm.
Is it a right-angled triangle? Give a reason.
OCR
Exam-style questions focusing on reasoning with the hypotenuse and mixed Pythagoras problems.
OCR
The hypotenuse is opposite the right angle.
Explain why the hypotenuse must always be the longest side.
OCR
A right-angled triangle has hypotenuse 17 cm and one shorter side 8 cm.
Find the other shorter side.
OCR
A ladder leans against a wall. The ladder is 13 m long and the foot of the ladder is 5 m from the wall.
Find the height h reached by the ladder.
Exam Checklist
Step 1
Check that the triangle is right-angled.
Step 2
Identify the hypotenuse correctly.
Step 3
Choose whether to add or subtract the squares.
Step 4
Take the square root at the end and include units if needed.
Most common exam mistakes
Wrong triangle
Using the theorem without a right angle.
Wrong hypotenuse
Not choosing the side opposite the right angle.
Wrong operation
Adding when you should subtract.
No square root
Stopping at \(c^2\) instead of finding \(c\).
Common Mistakes
These are common mistakes students make when using Pythagoras’ Theorem in GCSE Maths.
Using Pythagoras in non-right-angled triangles
Incorrect
A student applies the theorem to any triangle.
Correct
Pythagoras’ Theorem only works for right-angled triangles. Always check for a 90° angle before using it.
Choosing the wrong hypotenuse
Incorrect
A student identifies the wrong side as the hypotenuse.
Correct
The hypotenuse is always the longest side and is opposite the right angle. Label the triangle clearly before substituting.
Using the wrong operation
Incorrect
A student adds when they should subtract when finding a shorter side.
Correct
Use \(a^2 + b^2 = c^2\) when finding the hypotenuse. Rearrange to subtract (\(c^2 - a^2\)) when finding a shorter side.
Forgetting to square values
Incorrect
A student substitutes values without squaring them.
Correct
All sides must be squared before adding or subtracting. For example, use \(3^2\), not just 3.
Forgetting the square root
Incorrect
A student leaves the answer squared.
Correct
After finding the squared value of a side, take the square root to get the final answer.
Try It Yourself
Practise applying Pythagoras' theorem to right-angled triangles.
Foundation
Foundation Practice
Find missing sides in right-angled triangles using Pythagoras' theorem.
Find the hypotenuse of a right-angled triangle with shorter sides 5 cm and 12 cm.
Use a² + b² = c².
5² + 12² = 25 + 144 = 169, so c = 13 cm.
Which side is the hypotenuse?
It is always opposite the right angle.
The hypotenuse is the longest side and is opposite the right angle.
Find x.
Find the hypotenuse.
6² + 8² = 36 + 64 = 100, so x = 10 cm.
Find the missing shorter side.
Use hypotenuse² − known side².
10² − 8² = 100 − 64 = 36, so x = 6 cm.
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other shorter side.
Subtract the squares.
13² − 5² = 169 − 25 = 144, so the missing side is 12 cm.
Which calculation finds the hypotenuse when the shorter sides are 7 cm and 24 cm?
Add the squares, then square root.
For the hypotenuse, use \(\sqrt{7^2 + 24^2}\).
Find the hypotenuse when the shorter sides are 9 cm and 12 cm.
Use 9² + 12².
81 + 144 = 225, so the hypotenuse is 15 cm.
A triangle has sides 6 cm, 8 cm and 10 cm. Is it right-angled?
Check 6² + 8² = 10².
36 + 64 = 100, so it is right-angled.
A triangle has sides 5 cm, 12 cm and 13 cm. Is it right-angled? Enter yes or no.
Check whether the two smaller squares add to the largest square.
5² + 12² = 13², so it is right-angled.
Higher
Higher Practice
Solve multi-step Pythagoras problems, including decimals, coordinates and 3D contexts.
Find x to 1 decimal place.
Use \(x = \sqrt{7^2 + 9^2}\).
x = √(49 + 81) = √130 = 11.4 cm to 1 decimal place.
Find the missing shorter side to 1 decimal place.
Subtract the square of the known shorter side from the square of the hypotenuse.
x = √(11² − 6²) = √85 = 9.2 cm to 1 decimal place.
Find the distance between the points (1, 2) and (7, 10).
Find the horizontal and vertical differences first.
Difference in x = 6 and difference in y = 8. Distance = √(6² + 8²) = 10.
A ladder is 13 m long and reaches 12 m up a wall. How far is the foot of the ladder from the wall?
The ladder is the hypotenuse.
13² − 12² = 169 − 144 = 25, so x = 5 m.
A rectangle is 8 cm by 15 cm. Find the diagonal.
The diagonal is the hypotenuse of a right-angled triangle.
8² + 15² = 64 + 225 = 289, so the diagonal is 17 cm.
A square has side length 10 cm. Find its diagonal to 1 decimal place.
Use \(d^2 = 10^2 + 10^2\).
d = √200 = 14.1 cm to 1 decimal place.
A right-angled triangle has hypotenuse 20 cm and one shorter side 16 cm. Find the other shorter side.
Use 20² − 16².
400 − 256 = 144, so the missing side is 12 cm.
Which triangle is right-angled?
Check whether the two smaller squares equal the largest square.
9² + 12² = 81 + 144 = 225 = 15².
Find the distance from (−3, 4) to (5, −2), to 1 decimal place.
Find differences in x and y.
x difference = 8 and y difference = 6. Distance = √(8² + 6²) = 10.
A cuboid is 3 cm by 4 cm by 12 cm. Find the space diagonal.
Use 3D Pythagoras: diagonal² = 3² + 4² + 12².
3² + 4² + 12² = 9 + 16 + 144 = 169, so diagonal = 13 cm.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is Pythagoras' theorem?
a² + b² = c² in a right-angled triangle.
Which side is c?
The hypotenuse, the longest side.
When can I use it?
Only in right-angled triangles.