A sector of a circle has a central angle of 120° and a radius of 7 cm. Another sector of the same circle has a central angle of 2π⁄3 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, using π radians = 180°.

Convert: 2π⁄3 radians = 2 × 180° ÷ 3 = 120°

Both sectors lie in the same circle (r = 7 cm) with the same 120° angle.
Each area = (120 ⁄ 360) × π × 7² = 49π⁄3 ≈ 51.3 cm²
Ratio = 49π⁄3 : 49π⁄3 = 1 : 1 → option (c)

Why the others are wrong

• (a)
  3:5 — 3 : 5 needs a second angle of 200° (120 × 5⁄3). But 2π⁄3 radians is exactly 120°, the same as the first sector.
• (b)
  2:3 — 2 : 3 needs a second angle of 180°, which is π radians. The second angle here is 2π⁄3 radians, only 120°.
• (d)
  4:5 — 4 : 5 needs a second angle of 150°, which is 5π⁄6 radians. The second angle here, 2π⁄3, is 120°, equal to the first.

In one circle, a sector's area is (θ ⁄ 360°) × πr², so sectors of the same circle are in the ratio of their angles.

Radians and degrees are two units for the same angle: π radians = 180°. Multiply radians by 180 ⁄ π to get degrees.

So 2π⁄3 radians is 120°, the two sectors are the same size, and the 7 cm radius cancels from the ratio.

Key facts

• π radians = 180°, so 2π⁄3 radians = 120°.
• Area of a sector = (θ ⁄ 360°) × πr², or ½r²θ with θ in radians.
• Sectors of the same circle have areas in the ratio of their central angles.

Common traps

• Comparing 120 with 2π⁄3 ≈ 2.09 as bare numbers, without converting the units.
• Converting the wrong way: degrees to radians multiplies by π ⁄ 180, radians to degrees by 180 ⁄ π.

No directly related past PYQ was found.