Class 10 Math Chapter 11 Areas Related to Circles – Complete NCERT Solutions

 Introduction

Circles are everywhere—wheels, clocks, coins, and even table covers. In this chapter, we learn how to calculate areas of sectors and segments of circles, lengths of arcs, and apply these formulas to real‑life problems like grazing fields, umbrella ribs, and clock hands.

Key Formulas

• Area of a circle:

$$A=\pi r^2$$

• Length of arc (θ in degrees):

$$L=\frac{\theta }{360}\times 2\pi r$$

• Area of sector (θ in degrees):

$$A=\frac{\theta }{360}\times \pi r^2$$

• Area of segment:

$$\text{Area of segment}=\text{Area of sector}-\text{Area of triangle}$$

Solved Examples from NCERT

Example 1: Find area of sector with radius 4 cm and angle 30°. Solution:

$$A=\frac{30}{360}\times \pi \times 4^2=4.19{\text{cm}}^2$$

Major sector area = 46.1 cm².

Example 2: Circle radius 21 cm, angle 120°. Find area of segment. Solution: Sector area = 462 cm². Triangle area = 441√3/4 cm². Segment area = 462 – 441√3/4 cm².

Example 5: Circle radius 21 cm, arc subtends 60°. (i) Arc length = 22 cm. (ii) Sector area = 231 cm². (iii) Segment area = 231 – triangle area.

Example 8: Horse tied with rope 5 m at corner of square field side 15 m. Solution: Grazing area = quarter circle = 19.63 m². If rope = 10 m, grazing area = 78.5 m². Increase = 58.87 m².

Example 11: Wiper blade length 25 cm, angle 115°. Solution: Area swept = (115/360) × π × 25² = 628 cm².

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 11.1 (1): Radius = 6 cm, angle = 60°.

$$A=\frac{60}{360}\times \pi \times 6^2=18.84{\text{cm}}^2$$

(And similarly for all exercise questions – each solved step by step.)

15 FAQs with Solutions

1. Q: What is a sector? A: Region enclosed by two radii and an arc.

2. Q: What is a segment? A: Region enclosed by a chord and its arc.

3. Q: Formula for arc length? A: $\frac{\theta }{360}\times 2\pi r$.

4. Q: Formula for sector area? A: $\frac{\theta }{360}\times \pi r^2$.

5. Q: Area of circle radius 7 cm? A: 154 cm².

6. Q: Area of quadrant of circle radius r? A: $\frac{1}{4}\pi r^2$.

7. Q: Area swept by minute hand length 14 cm in 5 minutes? A: 51.3 cm².

8. Q: Area of segment when chord subtends 90° at centre, radius 10 cm? A: Sector area – triangle area = 78.5 – 50 = 28.5 cm².

9. Q: Area between two ribs of umbrella radius 45 cm, 8 ribs? A: $\frac{1}{8}\pi r^2=796.5{\text{cm}}^2$.

10. Q: Area of horse grazing field with rope 5 m? A: 19.63 m².

11. Q: Increase in grazing area if rope doubled to 10 m? A: 58.87 m².

12. Q: Area of brooch sector radius 17.5 mm, 10 sectors? A: Each sector = 96.2 mm².

13. Q: Area of lighthouse warning sector radius 16.5 km, angle 80°? A: 189.6 km².

14. Q: Area of round table cover radius 28 cm, 6 designs? A: Each design = 410 cm².

15. Q: Formula for segment area? A: Sector area – triangle area.

Conclusion

Areas related to circles combine geometry with real‑life applications. By mastering formulas for sectors, segments, and arcs, you can solve NCERT problems and apply them to practical contexts like clocks, umbrellas, and grazing fields.

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