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Geometry & Measures · Higher

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.

Revising 3d trigonometry & pythagoras the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “3D trigonometry & Pythagoras GCSE” on the site for the complete guide.

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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

  1. d = √(6² + 4² + 3²)
  2. d = √(36 + 16 + 9) = √61
  3. 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

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