GCSE · GCSE Maths
GCSE Maths Pythagoras' theorem
What is pythagoras' theorem?
Pythagoras' theorem states that in a right-angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides: a² + b² = c². It only works for right-angled triangles. GCSE Maths extends it to 3D problems (finding the body diagonal of a cuboid), distance between coordinates on a grid, and isosceles triangles split into two right-angled halves.
Worked example
A right-angled triangle has the two shorter sides 6 cm and 8 cm. Find the hypotenuse.
- Identify the hypotenuse: it's the side opposite the right angle and the longest side.
- Write Pythagoras: a² + b² = c² → 6² + 8² = c².
- Calculate: 36 + 64 = 100, so c² = 100.
- Square root: c = √100 = 10 cm.
Finding a shorter side (not the hypotenuse)
When you know the hypotenuse and one shorter side, rearrange: a² = c² − b². Don't confuse this with a² + b² = c² — getting the rearrangement wrong is the most common GCSE error on this topic. Example: hypotenuse 13, one side 5 → other side = √(13² − 5²) = √(169 − 25) = √144 = 12.
3D Pythagoras
For a cuboid with sides a, b, c, the body diagonal d satisfies d² = a² + b² + c². Derived by applying Pythagoras twice — first to find the face diagonal, then to find the body diagonal. Higher-tier GCSE examines this routinely.
Distance between two coordinates
Given points (x₁, y₁) and (x₂, y₂), the distance between them is √((x₂ − x₁)² + (y₂ − y₁)²) — Pythagoras applied to the horizontal and vertical distances. Examined on every UK GCSE Maths Higher paper.
Where it appears in the exam
AQA 8300, Edexcel 1MA1 and OCR J560 all examine Pythagoras on Foundation tier (basic right-angled triangle) and Higher tier (3D, coordinates, surd answers). Foundation typically asks for decimal answers; Higher often asks for 'exact form' which requires leaving the answer as a surd.
Common mistakes
- Squaring the hypotenuse on the wrong side of the equation when finding a shorter side.
- Forgetting to take the square root at the end.
- Rounding intermediate values too early — keep accuracy until the final answer.
- Trying to use Pythagoras on non-right-angled triangles (use sine rule or cosine rule instead).
Frequently asked
- How do I know which side is the hypotenuse?
- The hypotenuse is always opposite the right angle and is the longest side of a right-angled triangle. The right angle is marked with a small square in the corner.
- Can I use Pythagoras on any triangle?
- No — only right-angled triangles. For non-right-angled triangles, use the sine rule (a/sin A = b/sin B = c/sin C) or cosine rule (a² = b² + c² − 2bc·cosA).
- How is Pythagoras connected to trigonometry?
- SOHCAHTOA gives you sides or angles in a right-angled triangle when you know one angle and one side. Pythagoras gives you the third side when you know the other two. They are complementary — most exam questions use both together.
GCSE Maths glossary terms
- Pythagoras' theoremPythagoras' theorem states that in a right-angled triangle, the square on the hypotenuse equals the sum of squares on the other two sides: a² + b² = c². It applies only to right-angled triangles. GCSE questions extend it to 3D problems, isosceles triangles split into two right-angled halves, and finding distances between coordinates.
- SOHCAHTOASOHCAHTOA is the mnemonic for the three right-angled trig ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Use it in any right-angled triangle to find an unknown side or angle when you know one side + one angle (other than the right angle). For non-right-angled triangles, use the sine rule or cosine rule instead.
- Sine ruleThe sine rule relates the sides and opposite angles of any triangle: a / sin A = b / sin B = c / sin C. Use it when you know two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA, the ambiguous case). For right-angled triangles, SOHCAHTOA is simpler and preferred.
- Cosine ruleThe cosine rule relates the three sides and one angle of any triangle: a² = b² + c² − 2bc · cos A. Use it when you know two sides and the included angle (SAS), or all three sides and want to find an angle (SSS). For right-angled triangles, Pythagoras' theorem (cos 90° = 0) is the natural special case.
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