A ring-shaped metallic sheet has outer and inner radii 10 cm and 6 cm. What percentage of the full circle's area is the ring?

Correct — C. The full circle is the one with the outer radius, 10 cm.

Full circle = π × 10² = 100π
Hole = π × 6² = 36π
Ring = 100π − 36π = 64π
Ring ÷ full circle = 64π ⁄ 100π = 64% → option (c)

Why the others are wrong

• (a)
  36% — 36% is the hole, not the ring: 6² ⁄ 10² = 36⁄100. The ring is the rest of the circle, 100% − 36% = 64%.
• (b)
  48% — 48% would leave the hole at 52% of the circle. The hole is 6² ⁄ 10² = 36%, so the ring is 64%, not 48%.
• (d)
  76% — 76% would leave the hole at only 24% of the circle. The hole is 36⁄100 of the area, which leaves 64% for the ring.

A ring (annulus) is a large circle with a smaller, concentric circle cut out. Its area is π(R² − r²).

Compared with the full circle, π cancels, leaving (R² − r²) ⁄ R² = (100 − 36) ⁄ 100.

The inner radius is 60% of the outer, but the hole covers only 36% of the area, because area follows the square of the radius.

Key facts

• Area of a ring = π(R² − r²) = π(R + r)(R − r).
• Here π(R + r)(R − r) = π × 16 × 4 = 64π.
• Area scales with the square of the radius, so radii of 6 : 10 give areas of 36 : 100.

Common traps

• Subtracting the radii before squaring: (10 − 6)² = 16 gives 16%
• Reporting the hole's 36% instead of the ring's share
• Using the radius ratio 6⁄10 as though it were the area ratio

No directly related past PYQ was found.