Everything you need to know about Pythagoras' theorem for GCSE Maths. Find missing sides, check right angles, and apply to 3D problems with worked examples.
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Written as a formula: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).
Square the two shorter sides, add them together, then take the square root.
A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
Rearrange the formula: a² = c² − b². Square the hypotenuse, subtract the square of the known side, then take the square root.
A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
Pythagorean triples are sets of three whole numbers that satisfy the theorem. Recognising them speeds up exams: 3-4-5, 5-12-13, 8-15-17, and any multiples of these.
In Higher tier, you may need to find the length of a diagonal in a 3D shape (e.g., a cuboid or pyramid). Apply Pythagoras twice: first find a 2D diagonal, then use it as a leg to find the 3D diagonal.
A cuboid is 3 cm × 4 cm × 12 cm. Find the space diagonal.
No — only right-angled triangles. For other triangles, use the cosine rule: a² = b² + c² − 2bc cos A (Higher tier).
The hypotenuse is always opposite the right angle (90°), and it's always the longest side of the triangle.
Leave as a surd (e.g., √50 or 5√2) when asked for an exact answer, or when the question says "leave in simplest form". Round to decimal places only when the question asks you to.
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