A ring-shaped disc has outer radius 10 cm and inner radius 7 cm. What is the approximate ratio of the ring’s area to the whole outer circle?

Correct — A. The ring is the outer circle minus the inner circle.

Outer circle area = π × 10² = 100π
Inner circle area = π × 7² = 49π
Ring area = 100π − 49π = 51π

Ring : outer circle = 51π : 100π = 51 : 100
51 : 100 ≈ 50 : 100 = 1 : 2 → option (a)

Why the others are wrong

• (b)
  2:3 — 2 : 3 would need the ring to be about 67% of the outer circle, roughly 66.7π. The actual ring is 51π, which is 51%.
• (c)
  3:4 — 3 : 4 means a 75π ring. That needs an inner circle of 25π, a radius of 5 cm — not the 7 cm given.
• (d)
  4:5 — 4 : 5 means an 80π ring, which leaves an inner circle of just 20π, a radius of about 4.5 cm. A 7 cm inner radius removes 49π and leaves 51%.

A ring (annulus) is the region between two concentric circles. Its area is the outer area minus the inner area: π(R² − r²).

Both areas carry π, so the ratio of ring to outer circle is (R² − r²) : R². π cancels, and no value of it is needed.

Areas scale with the square of the radius: an inner radius of 7 out of 10 takes away 49% of the outer circle, not 70%.

Key facts

• Area of a ring = π(R² − r²) = π(R + r)(R − r).
• For R = 10 cm and r = 7 cm, the ring area is 51π ≈ 160.2 cm².
• Area ratios of circles are the squares of their radius ratios: 7 : 10 in radius is 49 : 100 in area.

Common traps

• Squaring the difference of the radii, (10 − 7)² = 9, instead of taking the difference of the squares, 100 − 49 = 51.
• Comparing the inner circle with the outer one. Here 49 : 100 also rounds to 1 : 2, but it answers a different question and fails when the radii change.

No directly related past PYQ was found.