GCSE Maths • Foundation & Higher
GCSE Pythagoras Revision Guide
Find missing lengths in right-angled triangles and 3D shapes.
Foundation
Exam practice
2D and 3D Pythagoras
What You'll Learn
Identify the Hypotenuse
Find the longest side opposite the right angle before choosing the correct rearrangement.
Find the Hypotenuse
Square both shorter sides, add, then take the positive square root.
Find a Shorter Side
Subtract the known shorter-side square from the hypotenuse square before square-rooting.
Use Pythagoras with Coordinates
Form a right-angled triangle from horizontal and vertical differences to calculate distance.
Work with Exact and Rounded Answers
Keep surds exact when required and round only the final value to the stated accuracy.
Solve 3D Problems
Identify two linked right-angled triangles to find face and space diagonals.
Key Pythagoras Rules and Formulas
Pythagoras Lessons
Basic Pythagoras
Identify the hypotenuse and find missing sides in right-angled triangles using clear step-by-step methods.
3d Pythagoras
Use linked right-angled triangles to calculate face diagonals and distances through cuboids and other 3D shapes.
Core Subtopics
Pythagorean Triples
Recognise integer side-length sets such as 3-4-5 and their multiples as useful checks.
Word Problems
Translate ladders, rectangles and shortest-distance contexts into accurately labelled right-angled triangles.
Distance Between Coordinates
Use coordinate differences as perpendicular lengths without relying on the visual scale of a grid.
Exact Surd Lengths
Leave non-square roots such as √50 in exact simplified form when the question requests an exact answer.
Repeated Pythagoras
Use an intermediate diagonal to create a second right-angled triangle in compound or 3D shapes.
Checking Triangle Type
Compare the square of the longest side with the sum of the other two squares.
Worked Example
Space Diagonal of a Cuboid
QUESTION
A cuboid measures 5 cm by 12 cm by 9 cm. Calculate the distance between two opposite vertices through the cuboid. Give the exact answer and a decimal answer to 3 significant figures.
- Given information: The three dimensions are mutually perpendicular. The required line is the space diagonal.
- Method: Find a face diagonal, then use it with the third perpendicular dimension.
- Find the base diagonal: p² = 5² + 12² = 25 + 144 = 169, so p = 13 cm.
- Form the second right-angled triangle using p = 13 cm and height 9 cm.
- Calculate the space diagonal: d² = 13² + 9² = 169 + 81 = 250.
- Take the positive square root: d = √250 = 5√10 cm.
- Round only at the end: 5√10 = 15.811… cm, so d = 15.8 cm to 3 significant figures.
- Exact distance = 5√10 cm; decimal distance = 15.8 cm to 3 significant figures.
- Using the combined cuboid formula gives d² = 5² + 12² + 9² = 25 + 144 + 81 = 250, matching the two-stage result.
- Do not round the 13 cm face diagonal unnecessarily or use a visible cuboid edge as the required diagonal. Identify both right-angled triangles first.
- Common GCSE Mistakes
- Using the Theorem Without a Right Angle
- Choosing the Wrong Hypotenuse
- Adding When Finding a Shorter Side
- Forgetting the Final Square Root
- Rounding Too Early
- Using the Wrong 3D Triangle
- Exam Tips
- Mark the right angle and label the hypotenuse before writing any formula.
- Write a² + b² = c² first, then substitute each length into the correct position.
- Keep units throughout and give the final length in the units requested.
- Check that the hypotenuse answer is longer than either shorter side; a shorter-side answer must be less than c.
- 25. For 3D questions, highlight each right-angled triangle and avoid rounding the intermediate diagonal.
Ready to Practice?
Printable Worksheet
Practise identifying hypotenuses, finding missing sides, coordinate distances and 3D diagonals.
Interactive Quiz
Check theorem conditions, formula rearrangements, Pythagorean triples, exact answers and common misconceptions.
Exam-Style Questions
Apply Pythagoras to diagrams, coordinates, word problems and multi-step Higher Tier contexts.
Watch the Pythagoras Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the two shorter sides. Using c for the hypotenuse, the theorem is a² + b² = c².
Find the side directly opposite the 90° angle. It is also the longest side of a right-angled triangle. Its position may change when a diagram is rotated, so do not identify it by appearance alone.
Start with a² + b² = c² and identify c as the hypotenuse. Subtract the square of the known shorter side from c², then take the positive square root. The result must be less than c.
Both tiers use Pythagoras in right-angled triangles and practical contexts. Higher Tier may include the converse, algebraic lengths, exact surd answers, coordinate geometry and repeated applications in 3D. Follow the selected examination board specification for exact coverage.
The equality a² + b² = c² is specific to right-angled triangles. For a non-right-angled triangle, other methods such as the cosine rule may be needed. A diagram that merely looks right-angled is not sufficient evidence.
Substitute the calculated length back into the squared equation. Check that the hypotenuse is the longest side, that units are correct and that any rounding matches the question. For a 3D diagonal, verify the result with l² + w² + h².
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