A sector of a circle has a central angle of 120° and a radius of 7 cm. Another…
2025
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?
Answer: C. 1 : 1 — Concept — A sector of radius r subtending a central angle θ has area ½r2θ when θ is measured in radians, and (θ/360°) × πr2 when θ is measured in degrees.…
- A.
3 : 5
- B.
2 : 3
- C.
1 : 1
- D.
4 : 5
Show answer & explanation
Correct answer: C
Concept — A sector of radius r subtending a central angle θ has area ½r2θ when θ is measured in radians, and (θ/360°) × πr2 when θ is measured in degrees. When two sectors belong to the same circle, the radius is common to both, so it cancels out of the comparison and the two areas stand in exactly the same ratio as their central angles. Two angles can be compared only after both are expressed in the same unit, using the conversion π radians = 180°.
Application
First sector: central angle = 120°.
Second sector: central angle = 2π/3 radians. Converting to degrees: (2π/3) × (180°/π) = (2 × 180°)/3 = 120°.
Both sectors lie in the same circle, so the radius 7 cm is common to both and cancels. Ratio of areas = ratio of central angles = 120° : 120°.
Ratio of the first sector's area to the second sector's area = 1 : 1.
Cross-check — Evaluating the two areas separately with r = 7 cm: first sector = (120°/360°) × π × 72 = 49π/3 cm2, second sector = ½ × 72 × (2π/3) = 49π/3 cm2, so the two areas come out identical and the ratio is 1 : 1. Note that the radius never enters the answer — it cancels — and that treating the second angle as the bare number 2π/3 ≈ 2.09 without first converting the units is what makes an unequal ratio look plausible.