IGCSE Maths / Pythagoras theorem
Pythagoras in three dimensions revision notes
Edexcel International GCSE Maths (4MA1), chapter 15: Pythagoras theorem, specification point 4.8d. Free lesson.
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Pythagoras in three dimensions means finding a right-angled triangle inside a solid and using Pythagoras’ theorem twice. First find a diagonal across the base, then use it in a second triangle to find the long diagonal through the solid.
Find the flat triangle in the base
A cuboid has rectangular faces, so its base contains a right-angled triangle. Redraw that triangle on its own before calculating. The diagonal across the base is the hypotenuse, the side opposite the right angle, so square the two shorter sides and add them. For a cuboid with AB = 8 cm and BC = 6 cm, triangle ABC is flat in the base. AC² = AB² + BC², so AC² = 8² + 6² = 64 + 36 = 100. Therefore AC = 10 cm. This is the diagonal across the base.
Stand a second triangle on the base diagonal
The long diagonal AG is found using a new triangle, ACG. The base diagonal AC lies flat and CG goes vertically up the cuboid, so they meet at a right angle at C. The side used from the first calculation is now one of the shorter sides. AG² = AC² + CG² = 10² + 5² = 125. AG = √125 ≈ 11.2 cm. For the angle GAC, use tangent: tan(GAC) = CG ÷ AC = 5 ÷ 10, so GAC ≈ 26.6°.
Use the correct length in the second triangle
The upright triangle ACG does not contain the edge AB. Its flat side is AC, the diagonal across the base. Using an edge instead of that diagonal gives the wrong result, even though the edge is easy to see in the cuboid. The incorrect calculation 8² + 5² = 89 uses AB instead of AC. The correct calculation is 10² + 5² = 125, because AC = 10 cm.
Work out a base diagonal on its own
If a question asks only for the diagonal across the base, ignore the height. Use the two measurements of the base, since those are the sides of the flat right-angled triangle. For a base measuring 12 cm by 9 cm, the diagonal is √(12² + 9²) = √225 = 15 cm. The cuboid’s height is not part of this triangle.
Keep the two calculations accurate
The method is always to find and redraw the first triangle, calculate its diagonal, then use that answer as a side in the upright triangle. Keep the full value between calculations and round only at the end, so rounding does not reduce the accuracy of your final answer.