3D trigonometry & Pythagoras

Problems in 3D require you to identify right-angled triangles within 3D shapes and apply Pythagoras' theorem or trigonometry to find lengths and angles.

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Key facts to remember

1Always identify a right-angled triangle within the 3D shape before applying formulas.
2To find the length of a space diagonal in a cuboid: apply Pythagoras twice (or use √(l² + w² + h²)).
3To find angles in 3D: identify a 2D right-angled triangle, then use SOH-CAH-TOA.
4The angle of elevation or depression in 3D problems uses the same trigonometric ratios.
5Draw and label the relevant 2D triangle extracted from the 3D shape.

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Formulas

Space diagonal of cuboid

d = √(l² + w² + h²)

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Worked examples

Example 1

A cuboid is 6 cm × 4 cm × 3 cm. Find the length of the space diagonal.

Working

d = √(6² + 4² + 3²)
d = √(36 + 16 + 9) = √61
d ≈ 7.81 cm

Answer√61 ≈ 7.81 cm

Example 2

A pyramid has a square base of side 8 cm and height 6 cm. Find the angle between the slant edge and the base.

Working

The diagonal of the base = √(8² + 8²) = √128 = 8√2 cm
Half the base diagonal = 4√2 cm
tan θ = 6 / (4√2) = 6 / 5.657 = 1.0607
θ = tan⁻¹(1.0607) ≈ 46.7°

Answer≈ 46.7°

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Common mistakes

✗Not identifying the correct right-angled triangle within the 3D shape.

✗Using the wrong dimensions — confusing height with slant height.

✗Not extracting and drawing the 2D triangle before applying trigonometry.

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Exam tips

✓Always draw the 2D triangle you are working with, labelled with known measurements.

✓Apply Pythagoras step by step: find a 2D diagonal first, then use it in a second triangle.

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