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 \…

Question

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 \(2π3\) radians. What is the ratio of the area of the first sector to the area of the second sector?

1. A. 3 : 5
2. B. 2 : 3
3. C. 1 : 1 (Correct answer)
4. D. 4 : 5

Correct Answer

Option C — 1 : 1

Detailed Solution & Explanation

The correct answer is 1 : 1.

Key Points

• Convert the second angle to degrees before comparing: 2π/3 radians = 120°, since π radians = 180°.
• Both sectors therefore subtend 120° and belong to the same circle, so they have the same radius as well.
• Sector area = (θ/360) × πr², and with θ and r identical for both, the areas are equal — the ratio is 1 : 1.
• No arithmetic with r = 7 cm is needed at all; the radius cancels.
• The question tests only the radian-to-degree conversion. Keep the anchors handy: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, 2π/3 = 120°, π = 180°.

Additional Information

• Radian-degree conversion: π radians = 180°, so degrees = radians × 180/π and radians = degrees × π/180.
• Sector formulae in both systems: area = (θ/360) × πr² with θ in degrees, or ½ r²θ with θ in radians; arc length = (θ/360) × 2πr, or rθ in radians.
• When two quantities share every parameter, their ratio is 1 : 1 and no computation is needed — spotting that saves the whole calculation.

प्रश्न (हिन्दी में)

एक वृत्त के एक त्रिज्यखंड का केंद्रीय कोण 120° और त्रिज्या 7 सेमी है। उसी वृत्त के एक अन्य त्रिज्यखंड का केंद्रीय कोण \(2π3\) रेडियन है। पहले त्रिज्यखंड के क्षेत्रफल का दूसरे त्रिज्यखंड के क्षेत्रफल से अनुपात क्या है?

1. A. 3 : 5
2. B. 2 : 3
3. C. 1 : 1
4. D. 4 : 5