A sector has a central angle of 180° and a radius of 12 cm. Another sector has a central angle of π⁄2 radians and the same radius. What is the ratio of the area of the first sector to the area of the second sector?

Correct — B.
Convert: π rad = 180°, so π⁄2 rad = 90°
First sector: (180⁄360) × π × 12² = 72π cm²
Second sector: (90⁄360) × π × 12² = 36π cm²

Ratio: 72π : 36π = 2 : 1 → option (b)
Shortcut: same radius, so the areas follow the angles, 180° : 90°

Why the others are wrong

• (a)
  1:2 — 1 : 2 is the ratio the wrong way round. The first sector (180°) is the larger one, so the first term must be the bigger.
• (c)
  3:1 — 3 : 1 would need the second angle to be 60° (π⁄3 rad). π⁄2 rad is 90°, and 180° : 90° = 2 : 1.
• (d)
  1:3 — 1 : 3 puts the smaller number first, yet the 180° sector is the bigger one. The angles give 2 : 1 in any case, not 3 : 1.

A sector's area is (θ⁄360°) × πr², or ½r²θ with θ in radians. For two sectors of the same radius, everything but θ cancels, so the areas are in the ratio of the angles.

The one thing to do first is put both angles in the same unit: π rad = 180°.

Key facts

• π radians = 180°, so π⁄2 rad = 90° and π⁄3 rad = 60°.
• Sector area = (θ⁄360°) × πr² = ½r²θ (θ in radians).
• Sectors of equal radius have areas in the ratio of their central angles.

Common traps

• Comparing 180 with π⁄2 as raw numbers without converting units.
• Writing the ratio second-to-first when the question asks first-to-second.

No directly related past PYQ was found.