The diameter of the moon is approximately one fourth the diameter of the earth. Given that the moon and earth are both roughly spherical, find the ratio of their surface areas.

Step-by-step solution

Idea: Surface area 4πr² depends on r², so the ratio of areas is the square of the ratio of radii (diameters).

1. Let the earth’s radius be r; then the moon’s radius is r4.½ mark
2. SA of moonSA of earth = 4π(r/4)²4πr² = 116.1 mark
3. Ratio = 1 : 16.½ mark

Check: Real radii ≈ 1737 km and 6371 km: (1737 ÷ 6371)² ≈ 0.074 ≈ 113.5, so 1 : 16 is a fair rough figure ✓.

Answer to write in the exam

Common mistakes that cost marks

• Answering 1 : 4 (not squaring).
• Answering 1 : 64 (cubing, which is for volumes).

How this can come in the exam

Try one yourself

The radii of two spheres are in the ratio 2 : 5. Find the ratio of their surface areas and of their volumes.

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