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?

Question

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?

1. A. 36%
2. B. 48%
3. C. 64% (Correct answer)
4. D. 76%

Correct Answer

Option C — 64%

Detailed Solution & Explanation

The correct answer is 64%.

Key Points

• The ring (annulus) is the outer circle minus the inner one:
  • Ring area = π(10² − 6²) = π(100 − 36) = 64π
• The full circle here is the outer one: π × 10² = 100π.
• Percentage = (64π ÷ 100π) × 100 = 64%.

Additional Information

• π cancels throughout, so the answer depends only on the squares of the radii — no value of π is needed and none should be substituted.
• General result: the annulus is (R² − r²)/R² of the outer circle, which here is (100 − 36)/100.
• The inner circle occupies the remaining 36%, and 64 + 36 = 100 ✓ — a quick check that the two parts account for the whole.
• The annulus formula generalises: for radii R and r the ring is π(R² − r²), which factorises to π(R + r)(R − r) — often quicker when the radii are close together.
• Here that gives π(10 + 6)(10 − 6) = π × 16 × 4 = 64π, confirming the result without squaring either radius.

प्रश्न (हिन्दी में)

एक वलय के आकार की धातु की शीट की बाहरी और भीतरी त्रिज्याएँ 10 सेमी और 6 सेमी हैं। वलय का क्षेत्रफल पूरे वृत्त के क्षेत्रफल का कितना प्रतिशत है?

1. A. 36%
2. B. 48%
3. C. 64%
4. D. 76%