A sector of a circle has a central angle of 120° and a radius of 5 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?
- (a)1:1
- (b)3:4
- (c)2:3
- (d)5:6
Answer
A
Why
Correct — A. Put both angles in the same unit, then compare.
Convert: 2π⁄3 rad × 180°⁄π = 120°
So both sectors have θ = 120° and r = 5 cm
Area of each = (120⁄360) × π × 5² = 25π⁄3 cm²
Ratio = 25π⁄3 : 25π⁄3 = 1 : 1 → option (a)
Why the others are wrong
- (b)3:4 — 3 : 4 would need the angles in the ratio 3 : 4, such as 120° and 160°. Here both angles are 120°.
- (c)2:3 — 2 : 3 would need the second angle to be 180° (π radians). The 2 and 3 in 2π⁄3 are not a ratio of the sectors.
- (d)5:6 — 5 : 6 would need angles such as 120° and 144°. The 5 cm radius is shared by both sectors and cancels, so it cannot shape the ratio.
Concept
A sector's area is (θ⁄360°) × πr² with θ in degrees, or ½r²θ with θ in radians. Both give 25π⁄3 cm² here.
In the same circle, r is fixed, so the areas of two sectors are in the ratio of their angles.
Convert with π radians = 180°: 2π⁄3 rad = 120°, π⁄3 rad = 60°, π⁄4 rad = 45°.
Key facts
- π radians = 180°, so 2π⁄3 radians = 120°.
- Sector area = (θ⁄360°) × πr² in degrees, or ½r²θ in radians.
- In the same circle, sector areas are proportional to their central angles.
Study next
Arc length of a sector (θ⁄360° × 2πr, or rθ in radians)Perimeter of a sector (2r + arc length)Degree–radian conversion for angles such as 5π⁄6 and 3π⁄4
Common traps
- Comparing 120 with 2π⁄3 ≈ 2.09 as if both were degrees.
- Letting the 5 cm radius into the ratio when both sectors share it.
12 Sep 2025, 16:00, Quant Q.25 has the same 120° and 2π⁄3 pair in a 7 cm circle, keyed 1:1.
17 Sep 2025, 16:00, Quant Q.17 changes the angles to 180° and π⁄2 radians at the same radius, keyed 2:1.
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