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# Class 10 RD Sharma Solutions – Chapter 15 Areas Related to Circles – Exercise 15.2

• Last Updated : 03 May, 2021

### Question 1. Find, in terms of Ï€, the length of the arc that subtends an angle of 30o at the centre of a circle of radius of 4 cm.

Solution:

Given,

Radius = 4 cm

Angle subtended at the centre = 30Â°

Length of arc = Î¸/360 Ã— 2Ï€r

Length of arc = 30/360 Ã— 2Ï€ Ã— 4 cm

= 2Ï€/3

Therefore, the length of arc that subtends an angle of 30o degree is 2Ï€/3 cm

### Question 2. Find the angle subtended at the centre of a circle of radius 5 cm by an arc of length 5Ï€/3 cm.

Solution:

Length of arc = 5Ï€/3 cm

Length of arc = Î¸/360 Ã— 2Ï€r cm

5Ï€/3 cm = Î¸/360 Ã— 2Ï€r cm

Î¸ = 60Â°

Therefore, the angle subtended at the centre of circle is 60Â°

### Question 3. An arc of length 20Ï€ cm subtends an angle of 144Â° at the centre of a circle. Find the radius of the circle.

Solution:

Length of arc = 20Ï€ cm

Î¸ = Angle subtended at the centre of circle = 144Â°

Length of arc = Î¸/360 Ã— 2Ï€r cm

Î¸/360 Ã— 2Ï€r cm = 144/360 Ã— 2Ï€r cm = 4Ï€/5 Ã— r cm

20Ï€ cm = 4Ï€/5 Ã— r cm

r = 25 cm.

Therefore, the radius of the circle is 25 cm.

### Question 4. An arc of length 15 cm subtends an angle of 45Â° at the centre of a circle. Find in terms of Ï€, the radius of the circle.

Solution:

Length of arc = 15 cm

Î¸ = Angle subtended at the centre of circle = 45Â°

Length of arc = Î¸/360 Ã— 2Ï€r cm

= 45/360 Ã— 2Ï€r cm

15 cm = 45/360 Ã— 2Ï€ Ã— r cm

15 = Ï€r/4

Radius = 15Ã—4/ Ï€ = 60/Ï€

Therefore, the radius of the circle is 60/Ï€ cm.

### Question 5. Find the angle subtended at the centre of a circle of radius â€˜aâ€™ cm by an arc of length (aÏ€/4) cm.

Solution:

Radius = a cm

Length of arc = aÏ€/4 cm

Î¸ = angle subtended at the centre of circle

Length of arc = Î¸/360 Ã— 2Ï€r cm

Î¸/360 Ã— 2Ï€a cm = aÏ€/4 cm

Î¸ = 360/ (2 x 4)

Î¸ = 45Â°

Therefore, the angle subtended at the centre of circle is 45Â°

### Question 6. A sector of a circle of radius 4 cm subtends an angle of 30Â°. Find the area of the sector.

Solution:

Radius = 4 cm

Angle subtended at the centre O = 30Â°

Area of the sector = Î¸/360 Ã— Ï€r2

= 30/360 Ã— Ï€42

= 1/12 Ã— Ï€16

= 4Ï€/3 cm

= 4.19 cm

Therefore, the area of the sector of the circle = 4.19 cm

### Question 7. A sector of a circle of radius 8 cm contains an angle of 135o. Find the area of sector.

Solution:

Radius = 8 cm

Angle subtended at the centre O = 135Â°

Area of the sector = Î¸/360 Ã— Ï€r2

Area of the sector = 135/360 Ã— Ï€82

= 24Ï€ cm2

= 75.42 cm2

Therefore, area of the sector calculated = 75.42 cm2

### Question 8. The area of a sector of a circle of radius 2 cm is Ï€ cm2. Find the angle contained by the sector.

Solution:

Radius = 2 cm

Area of sector of circle = Ï€ cm2

Area of the sector = Î¸/360 Ã— Ï€r2

= Î¸/360 Ã— Ï€22

= Ï€Î¸/90

Ï€  = Ï€ Î¸/90

Î¸ = 90Â°

Therefore, the angle subtended at the centre of circle is 90Â°

### Question 9. The area of a sector of a circle of radius 5 cm is 5Ï€ cm2. Find the angle contained by the sector.

Solution:

Radius = 5 cm

Area of sector of circle = 5Ï€ cm2

Area of the sector = Î¸/360 Ã— Ï€r2

= Î¸/360 Ã— Ï€52

= 5Ï€Î¸/72

5Ï€  = 5Ï€Î¸/72

Î¸ = 72Â°

Therefore, the angle subtended at the centre of circle is 72Â°

### Question 10. Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm.

Solution:

Radius = 5 cm

Length of arc = 3.5 cm

Length of arc = Î¸/360 Ã— 2Ï€r cm

= Î¸/360 Ã— 2Ï€(5)

3.5 = Î¸/360 Ã— 2Ï€(5)

3.5 = 10Ï€ Ã— Î¸/360

Î¸ = 360 x 3.5/ (10Ï€)

Î¸ = 126/ Ï€

Area of the sector = Î¸/360 Ã— Ï€r2

= (126/ Ï€)/ 360 Ã— Ï€(5)2

= 126 x 25 / 360

= 8.75

Therefore, the area of the sector = 8.75 cm2

### Question 11. In a circle of radius 35 cm, an arc subtends an angle of 72Â° at the centre. Find the length of the arc and area of the sector.

Solution:

Radius = 35 cm

Angle subtended at the centre = 72Â°

Length of arc = Î¸/360 Ã— 2Ï€r cm

= 72/360 Ã— 2Ï€(35)

= 14Ï€

= 14(22/7)

= 44 cm

Area of the sector = Î¸/360 Ã— Ï€r2

= 72/360 Ã— Ï€ 352

= (0.2) x (22/7) x 35 Ã— 35

= 0.2 Ã— 22 Ã— 5 Ã— 35

Area of the sector = (35 Ã— 22) = 770 cm2

Length of arc = 44cm

### Question 12. The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.

Solution:

Perimeter of sector includes length of arc and two radii

Radius = 5.7 cm = OA = OB

Perimeter of the sector = 27.2 m

Length of arc = Î¸/360 Ã— 2Ï€r m

Perimeter = l + 2r

Perimeter of the sector = Î¸/360 Ã— 2Ï€r + OA + OB

27.2 = Î¸/360 Ã— 2Ï€ x 5.7 cm + 5.7 + 5.7

27.2 â€“ 11.4 = Î¸/360 Ã— 2Ï€ x 5.7

15.8 = Î¸/360 Ã— 2Ï€ x 5.7

Î¸ = 158.8Â°

Area of the sector = Î¸/360 Ã— Ï€r2

Area of the sector = 158.8/360 Ã— Ï€ 5.72

Area of the sector = 45.03 m2

### Question 13. The perimeter of a certain sector of a circle of radius is 5.6 m and 27.2 m. Find the area of the sector.

Solution:

Radius of the circle = 5.6 m = OA = OB

Perimeter of the sector = Perimeter = l + 2r = 27.2

Length of arc = Î¸/360 Ã— 2Ï€r cm

Î¸/360 Ã— 2Ï€r cm + OA + OB = 27.2 m

Î¸/360 Ã— 2Ï€r cm + 5.6 + 5.6 = 27.2 m

Î¸ = 163.64Â°

Area of the sector = Î¸/360 Ã— Ï€r2

Area of the sector = 163.64/360 Ã— Ï€ 5.62

= 44.8

Therefore, the area of the sector = 44.8 m2

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