A sector has a central angle of 90° and a radius of 6 cm. Another sector has a central angle of π⁄2radians. What is the ratio of the area of the first sector to the area of the second sector?

Correct — B. Convert the radian angle to degrees, then compare the areas.

π radians = 180°
π⁄2 radians = 180° ÷ 2 = 90°

The stem gives no second radius. Taking it as 6 cm, as the key does:
Each area = (90⁄360) × π × 6² = 9π cm²
Ratio = 9π : 9π = 1 : 1 → option (b)

Why the others are wrong

• (a)
  2:1 — 2 : 1 would need the first angle to be double the second. But π⁄2 radians is exactly 90°, the same as the first angle.
• (c)
  3:2 — 3 : 2 needs one sector 1.5 times the other. Converted, π⁄2 radians = 90°, so at the same radius the two areas are equal.
• (d)
  5:6 — 5 : 6 makes the first sector smaller, yet its angle, 90°, equals π⁄2 radians. At the same radius, equal angles give equal areas.

A sector's area is the fraction of the full circle its angle covers: (θ⁄360°) × πr² with θ in degrees, or (1⁄2)r²θ with θ in radians.

Degrees and radians are two units for the same angle, linked by π rad = 180°. Once π⁄2 is seen as 90°, the two sectors have the same angle, and at equal radius their areas are equal.

Check in radians: (1⁄2) × 36 × π⁄2 = 9π, as the degree formula gives.

Key facts

• π radians = 180°, so π⁄2 = 90°, π⁄3 = 60° and π⁄4 = 45°.
• Sector area = (θ⁄360°) × πr², or (1⁄2)r²θ with θ in radians.
• At equal radius, sector areas are in the ratio of their angles.

Common traps

• Comparing 90 with π⁄2 ≈ 1.57 as if both were in the same unit

No directly related past PYQ was found.