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0580/31 May/Jun 2021, Question 8: Composite Perimeter, Area and Volume

Cambridge IGCSE Mathematics 0580 · Paper 31 (Core) · Calculator · 11 marks

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The question, in short

Use a circle with diameter AC and point B on it, where AB = 6 cm and BC = 10 cm, to justify angle ABC, find the triangle's area and AC, rearrange A = πr² for r, and find the shaded area between a circle and a square of area 81 cm².

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

Part (ai) · 1 mark

State why angle ABC equals 90°, referring to AC being a diameter.

  1. 1AC is a diameter, so angle ABC is 90° because it is the angle in a semicircle

AnswerAngle in a semicircle

Part (aii) · 2 marks

Calculate the area of triangle ABC using AB = 6 cm and BC = 10 cm.

  1. 1Angle ABC = 90°, since AC is a diameter (angle in a semicircle).
  2. 2So AB and BC are the base and height.
  3. 3Area = 1/2 × 6 × 10 = 30 cm²

Answer30

Part (aiii) · 2 marks

Find the length of AC.

  1. 1Angle ABC = 90°, since AC is a diameter, so use Pythagoras.
  2. 2AC² = 6² + 10² = 36 + 100 = 136
  3. 3AC = √136 = 11.66… ≈ 11.7 cm

Answer11.7 or 11.66…

Part (b) · 2 marks

Rearrange A = πr² to make r the subject.

  1. 1A/π = r²
  2. 2r = √(A/π)

Answer√(A/π)

Part (c) · 4 marks

The circle touches all sides of a square of area 81 cm²; find the shaded area between them.

  1. 1Side of square = √81 = 9 cm, so radius = 9 ÷ 2 = 4.5 cm
  2. 2Circle area = π × 4.5² = 63.61…
  3. 3Shaded area = 81 − 63.61…
  4. 4= 17.38… ≈ 17.4 cm²

Answer17.4 or 17.37 to 17.38…

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Also searched as: 0580/31/M/J/21 · 0580_s21_qp_31 · 0580 s21 31 · question 8.
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