Curriculum fit: GCSE Maths Geometry and measures (Foundation and Higher). US: Grade 8 and Geometry (Right Triangles and Trigonometry).
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What Is It?
Pythagoras' theorem links the three sides of a right-angled triangle. The longest side, opposite the right angle, is called the hypotenuse.
It lets you find a missing side when you know the other two, and it is used in many real-life problems such as ladders, ramps and distances on a grid.
Key Formulas and Ideas
- a² + b² = c², where c is the hypotenuse.
- To find the hypotenuse: add the squares of the two shorter sides, then take the square root.
- To find a shorter side: subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
- A triangle is right-angled if its sides satisfy a² + b² = c².
Worked Examples
Find the Hypotenuse
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
- c² = 6² + 8² = 36 + 64 = 100.
- c = √100 = 10.
✓ Answer 10 cm
Find a Shorter Side
The hypotenuse is 13 cm and one side is 5 cm. Find the other side.
- b² = 13² − 5² = 169 − 25 = 144.
- b = √144 = 12.
✓ Answer 12 cm
A Ladder
A 5 m ladder stands with its base 3 m from a wall. How high does it reach?
- The ladder is the hypotenuse: 5² = 25.
- Height² = 25 − 3² = 25 − 9 = 16.
- Height = √16 = 4.
✓ Answer 4 m
Common Mistakes to Avoid
- Adding when you need to find a shorter side, instead of subtracting.
- Forgetting to take the square root at the end.
- Using the theorem on triangles that are not right-angled.
- Choosing the wrong side as the hypotenuse.
Practice Questions
Frequently Asked Questions
Does it work for any triangle?
No. It only works for right-angled triangles.
What are Pythagorean triples?
Sets of whole numbers that satisfy the theorem, such as 3, 4, 5 and 5, 12, 13.
Keep Going With These Guides
SOH CAH TOA: How to Find Sides and Angles in a Triangle
Use sine, cosine and tangent to find missing sides and angles in right-angled triangles, with step-by-step examples for GCSE and high school Geometry.
GCSEHigh SchoolSine Rule and Cosine Rule: Formulas and Worked Examples
Solve non-right-angled triangles with the sine and cosine rules (law of sines and law of cosines) and find a triangle's area, with examples.
GCSEA-LevelHigh SchoolFind the Equation of a Line From Two Points
Work out the gradient (slope), find the y-intercept and write the equation of a line, plus parallel and perpendicular lines and midpoints.
GCSEA-LevelHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.