A sphere of radius 21 cm is cut into 8 identical pieces by making three mutually perpendicular cuts through its center (one along…

Question

A sphere of radius 21 cm is cut into 8 identical pieces by making three mutually perpendicular cuts through its center (one along each axis). Find the total surface area of each piece in cm².

1. A. 560π cm²
2. B. 551.25π cm² (Correct answer)
3. C. 450.75π cm²
4. D. 625.25π cm²

Correct Answer

Option B — 551.25π cm²

Detailed Solution & Explanation

The correct answer is 551.25π cm².

Key Points

• Three mutually perpendicular cuts through the centre divide the sphere into 8 identical octants, each bounded by one curved surface and three flat quarter-circles.
• Curved part: one-eighth of the sphere's surface.
  • (1/8) × 4πr² = (1/2)π(21²) = 220.5π cm²
• Flat parts: each cut exposes a quarter of a great circle, and each piece has three.
  • 3 × (1/4)πr² = (3/4)π(441) = 330.75π cm²
• Total = 220.5π + 330.75π = 551.25π cm².

Additional Information

• The trap is counting only the curved surface. Cutting a solid creates new faces, and their area must be added — the same principle applies when a cylinder or cone is sliced.
• Sphere formulae: surface = 4πr², volume = (4/3)πr³; a hemisphere has curved surface 2πr² and total surface 3πr² once its flat face is counted.
• Each octant's volume is (1/8)(4/3)πr³ = (1/6)πr³, useful if the question asks for volume instead.
• The answer is left in terms of π to match the options, so no approximation is needed.