A sector of a circle with a radius of 15 cm has a central angle of 45°. Another sector of the same circle has a central angle of π⁄4 radians. What is the ratio of the area of the first sector to the area of the second sector?

Correct — C. Put both angles in the same unit before comparing.

Degrees in π radians: 180°
Convert π⁄4 radians: 180° ÷ 4 = 45°
First sector 45°, second sector 45°

Area of each: 45⁄360 × π × 15² = 225π⁄8 cm²
Ratio: 225π⁄8 : 225π⁄8 = 1:1 → option (c)

Why the others are wrong

• (a)
  1:2 — 1:2 would need the second angle to be 90°, which is π⁄2 radians. π⁄4 radians is 45°, the same as the first sector.
• (b)
  2:3 — 2:3 would need a second angle of 67.5°. In one circle, sector areas follow the ratio of their angles, and 45° : 45° is 1:1.
• (d)
  3:4 — 3:4 would need the second angle to be 60°, which is π⁄3 radians, not π⁄4.

In one circle, sector area is proportional to the central angle: area = θ⁄360 × πr² with θ in degrees, or ½r²θ with θ in radians. The shared 15 cm radius cancels.

What the question really tests is the unit change. π radians = 180°, so π⁄4 radians = 45°, and two 45° sectors of one circle have equal areas.

Key facts

• π radians = 180°, so 1 radian ≈ 57.3°.
• Sector area = θ⁄360 × πr² in degrees, or ½r²θ in radians.
• In one circle, sector areas are in the ratio of their central angles.

Common traps

• Mixing up π⁄4 with π⁄2 (90°), which gives 1:2, option (a).
• Working out both areas with π = 3.14 when the angles alone settle the ratio.

No directly related past PYQ was found.