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0580/42 Feb/Mar 2024, Question 4: Surface Area of a Pyramid

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 11 marks

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The question, in short

Pyramid with square base 12 cm sides and vertical height 9 cm; find its volume and total surface area, then find r for a cone-on-hemisphere toy with slant height 3r and total surface area 304 cm².

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Part (ai) · 2 marks

Square base side 12 cm, height 9 cm; use V = 1/3 Ah to calculate the pyramid's volume.

  1. 1V = 1/3 × 12 × 12 × 9
  2. 2= 432 cm³

Answer432

Part (aii) · 5 marks

Calculate the pyramid's total surface area, including the square base and four triangular faces.

  1. 1Height of each triangular face = √(6² + 9²) = √117 = 10.81… cm
  2. 2Area of one triangle = 1/2 × 12 × 10.81… = 64.89… cm²
  3. 3Total = 12² + 4 × 64.89…
  4. 4= 144 + 259.6… = 403.6… ≈ 404 cm²

Answer404 or 403.5 to 403.7

Part (b) · 4 marks

Cone radius r, slant height 3r, on hemisphere radius r; total surface area 304 cm²; calculate r.

  1. 1Hemisphere curved surface = 1/2 × 4πr² = 2πr²
  2. 2Cone curved surface = π × r × 3r = 3πr²
  3. 32πr² + 3πr² = 5πr² = 304
  4. 4r² = 304 ÷ (5π) = 19.35…
  5. 5r = 4.398… ≈ 4.40

Answer4.40 or 4.398 to 4.399…

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