If the area of a sector of a circle is 462 cm² and the central angle is 120°, then its radius is ____.

Correct — D. A 120° sector is 120⁄360 = one third of its circle, so a third of the circle's area is 462 cm².

(1⁄3) × (22⁄7) × r² = 462
r² = 462 × 3 × 7 ⁄ 22
r² = 441
r = √441 = 21 cm → option (d).

Check: the whole circle is (22⁄7) × 441 = 1386 cm², and a third of 1386 is 462.

Why the others are wrong

• (a)
  18 cm — 18 cm makes the sector (1⁄3)(22⁄7)(324) ≈ 339.4 cm², well under the 462 the question states.
• (b)
  17 cm — 17 cm gives (1⁄3)(22⁄7)(289) ≈ 302.8 cm². It is tempting because treating the sector as half the circle gives r ≈ 17.1, but 120° is a third, not a half.
• (c)
  14 cm — 14 cm is the radius carried over from the familiar circle of area (22⁄7)(196) = 616 cm². As a 120° sector it measures only about 205.3 cm².

A sector's area is the circle's area scaled by the fraction of the full turn it covers: (θ⁄360) × πr². At θ = 120° that fraction is exactly one third.

Working backwards to r, multiply both sides by 360⁄θ and then divide by π. Taking π = 22⁄7 here collapses 462 × 3 × 7 ⁄ 22 to 441, a perfect square — that clean landing is the signal that 22⁄7 was the intended value rather than 3.14.

Key facts

• Area of a sector = (θ⁄360) × πr².
• A 120° sector is exactly one third of its circle.
• If a third of a circle is 462 cm², the whole circle is 1386 cm², and (22⁄7)r² = 1386 gives r = 21.
• Arc length is (θ⁄360) × 2πr, a different formula from the area of the same sector.

Common traps

• Using the arc-length formula when the question supplies an area
• Dividing by 120 rather than scaling by 120⁄360
• Stopping at r² = 441 and forgetting the square root

No directly related past PYQ was found.