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4024/23 May/Jun 2025, Question 22: Volume of a Pyramid

Cambridge O Level Mathematics 4024 · Paper 23 (Calculator) · Calculator · 11 marks

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The question, in short

For a pyramid with equilateral base side 34 cm and equal sloping edges 82 cm, find symmetry planes, base distance OB, an angle at the base, and the volume.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

State how many planes of symmetry the pyramid has.

  1. 1One plane through V, each vertex of the base and the midpoint of the opposite side
  2. 2Number of planes = 3

Answer3

Part (bi) · 3 marks

Using M as midpoint of AC and MO:OB = 1:2, prove OB rounds to 19.6 cm.

  1. 1MB² = 34² − 17² = 1156 − 289 = 867
  2. 2MB = √867 = 29.44…
  3. 3OB = 2/3 × 29.44… = 19.62… ≈ 19.6 cm

Answer19.62 to 19.63

Part (bii) · 2 marks

Find the angle that edge VB makes with the base plane.

  1. 1M is the midpoint of AC, so BM = √(34² − 17²) = √867 = 29.44…
  2. 2MO : OB = 1 : 2, so OB = 2/3 × 29.44… = 19.63…
  3. 3Triangle VOB is right-angled: cos VBO = 19.63…/82
  4. 4Angle VBO = 76.15… ≈ 76.2°

Answer76.1 to 76.2

Part (biii) · 5 marks

Work out the pyramid's volume.

  1. 1VO = √(82² − 19.62…²) = 79.61…
  2. 2Base area = 1/2 × 34 × 34 × sin 60° = 500.56…
  3. 3Volume = 1/3 × 500.56… × 79.61…
  4. 4= 13284… ≈ 13300 cm³

Answer13300 or 13281 to 13286

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