A sector of a circle has a central angle of 60° and a radius of 9 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 — A.
Convert the second angle to degrees: π⁄3 × (180°⁄π) = 60°
Both sectors are in the same circle, so both have r = 9 cm.

Sector area = (θ⁄360°) × πr²
First: (60⁄360) × π × 81 = 13.5π cm²
Second: (60⁄360) × π × 81 = 13.5π cm²
Ratio = 13.5π : 13.5π = 1 : 1 → option (a).

Why the others are wrong

• (b)
  2:3 — 2 : 3 would need a second angle of 90° (π⁄2 radians), since with one radius the areas follow the angles: 60 : 90 = 2 : 3. Here it is 60°.
• (c)
  3:5 — 3 : 5 would need a second angle of 100° (5π⁄9 radians): 60 : 100 = 3 : 5. π⁄3 radians is 60°, not 100°.
• (d)
  5:6 — 5 : 6 would need a second angle of 72° (2π⁄5 radians): 60 : 72 = 5 : 6. π⁄3 radians converts to 60°, the same as the first.

π radians = 180°, so π⁄3 radians = 60°. The two angles are written in different units; converted, they are equal.

Sector area = (θ⁄360°) × πr², or ½r²θ with θ in radians. In one circle the radius is shared, so sector areas are in the ratio of the angles.

Key facts

• π radians = 180°, so 1 radian ≈ 57.3°.
• Sector area = (θ⁄360°) × πr² with θ in degrees, or ½r²θ with θ in radians.
• Arc length = (θ⁄360°) × 2πr, or rθ with θ in radians.
• In the same circle, sector areas are in the ratio of their central angles.

Common traps

• Comparing 60 with π⁄3 ≈ 1.05 as if both were in the same unit
• Treating the second sector's radius as unknown when "the same circle" already fixes it at 9 cm

No directly related past PYQ was found.