A sector of a circle has a central angle of 60° and a radius of 8 cm. Another sector of the same circle has a central angle of π/3 radians. What is the ratio of the area of the first sector to the area of the second sector?

Correct — C.
Convert the second angle: π⁄3 rad = 180° ÷ 3 = 60°

Same circle, so both sectors have r = 8 cm.
Area of a sector = (θ⁄360°) × πr²
First: (60⁄360) × π × 64 = 32π⁄3 cm²
Second: same angle, same radius = 32π⁄3 cm²

Ratio = 1:1 → option (c)

Why the others are wrong

• (a)
  2:3 — 2:3 would need the second angle to be 90° (π⁄2 rad). In one circle area follows the angle, and π⁄3 rad is 60°, the same as the first.
• (b)
  1:2 — 1:2 would need the second sector to have twice the angle, 120° or 2π⁄3 rad. π⁄3 rad is 60°, equal to the first angle.
• (d)
  1:3 — 1:3 would need a second angle of 180°, which is π rad. The given π⁄3 rad converts to 60°, equal to the first.

In one circle, a sector's area is proportional to its central angle: area = (θ⁄360°) × πr², or (1⁄2)r²θ with θ in radians. With r the same, the area ratio is the angle ratio.

So the question is a unit conversion: π rad = 180°, so π⁄3 rad = 60°. Two 60° sectors of the same circle have equal areas.

Key facts

• π radians = 180°, so π⁄3 rad = 60°.
• Area of a sector = (θ⁄360°) × πr², or (1⁄2)r²θ with θ in radians.
• Sectors of the same circle have areas in the ratio of their central angles.

Common traps

• Putting θ = 60 into (1⁄2)r²θ: that formula needs the angle in radians, π⁄3.
• Comparing 60 with π⁄3 ≈ 1.05 as if both were in the same unit.

No directly related past PYQ was found.