source: ncert_exercises
topic: geometry
class_level: class_10
chapter: ch11
exercise: ex11.1
difficulty: intermediate
Class 10 | Ch11: Areas Related to Circles | Exercise 11.1
Key Formulas (r=radius): Area of circle = πr² Circumference = 2πr Area of semicircle = πr²/2 Area of quadrant = πr²/4 Area of sector (angle θ°) = (θ/360)×πr² Length of arc = (θ/360)×2πr Area of segment = Area of sector - Area of triangle
Q1. Area of sector whose radius=7cm and angle=60°. Answer: Area=(60/360)×π×49=49π/6=49×22/(7×6)=77/3≈25.67cm². ✅
Q2. Find area of quadrant of circle whose circumference is 22cm. Answer: 2πr=22 → r=7/2=3.5cm. Area of quadrant=πr²/4=π×12.25/4=22/7×12.25/4=9.625cm². ✅
Q3. Length of minute hand of clock=14cm. Find area swept in 5 minutes. Answer: In 60 min, minute hand sweeps 360°. In 5 min: 30°. Area=(30/360)×π×196=196π/12=154/3≈51.3cm². ✅
Q4. Chord of circle radius 10cm subtends 90° at centre. Find: (i) area of minor sector (ii) area of major sector (iii) area of minor segment (iv) area of major segment Answer: r=10, θ=90°. (i) Area of sector=90/360×π×100=25π=78.5cm². ✅ (ii) Area of major sector=270/360×π×100=75π=235.5cm². ✅ (iii) Area of triangle=½×10×10=50cm². Area of minor segment=78.5-50=28.5cm². ✅ (iv) Area of major segment=π×100-28.5=285.5cm². ✅
Q5. In a circle of radius 21cm, an arc subtends 60° at centre. Find: (i) length of arc (ii) area of sector (iii) area of segment Answer: r=21, θ=60°. (i) Arc=60/360×2π×21=22cm. ✅ (ii) Sector=60/360×π×441=231cm². ✅ (iii) For 60° in circle of radius 21: triangle is equilateral (chord=radius=21). Area of △=√3/4×441=110.25√3≈190.9cm²... Wait r=21,θ=60°: triangle area=½r²sinθ=½×441×sin60°=220.5×√3/2≈190.9cm². Segment=231-190.9=40.1cm². Actually: Area=½×21²×sin60°=441√3/4=190.9. Segment=231-190.9=40.1cm². ✅
Q6. A chord of circle of radius 15cm subtends angle of 60°. Find areas of minor segment and major sector. Answer: r=15, θ=60°. Area sector=60/360×π×225=37.5π=117.75cm². Triangle area=½×225×sin60°=225√3/4=97.3cm². Minor segment=117.75-97.3=20.4cm². Major sector=5/6×π×225=589cm². ✅