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

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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 = 30°

Length of arc = θ/360 × 2Ï€r 

Length of arc = 30/360 × 2Ï€ × 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 = θ/360 × 2πr cm

5Ï€/3 cm = θ/360 × 2Ï€r cm

θ = 60°

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

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

Solution:

Length of arc = 20Ï€ cm

θ = Angle subtended at the centre of circle = 144°

Length of arc = Î¸/360 × 2Ï€r cm

θ/360 × 2Ï€r cm = 144/360 Ã— 2Ï€r cm = 4Ï€/5 × r cm

20Ï€ cm = 4Ï€/5 × 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 45° at the centre of a circle. Find in terms of Ï€, the radius of the circle. 

Solution:

Length of arc = 15 cm

θ = Angle subtended at the centre of circle = 45°

Length of arc = Î¸/360 × 2Ï€r cm

                = 45/360 × 2Ï€r cm

15 cm = 45/360 × 2π × r cm

15 = Ï€r/4

Radius = 15×4/ 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 ‘a’ cm by an arc of length (aÏ€/4) cm. 

Solution:

Radius = a cm

Length of arc = aÏ€/4 cm

θ = angle subtended at the centre of circle

Length of arc = Î¸/360 × 2Ï€r cm

θ/360 × 2Ï€a cm = aÏ€/4 cm

θ = 360/ (2 x 4)

θ = 45°

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

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

Solution:

Radius = 4 cm

Angle subtended at the centre O = 30°

Area of the sector = θ/360 × πr2

                             = 30/360 × Ï€42 

                             = 1/12 × 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 = 135°

Area of the sector = θ/360 × πr2

Area of the sector = 135/360 × π82

                             = 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 = π cm2

Area of the sector = θ/360 × πr2

                             = θ/360 × Ï€22

                            = Ï€Î¸/90

Ï€  = Ï€ θ/90

θ = 90°

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

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 = θ/360 × πr2

                             = θ/360 × Ï€52

                             = 5πθ/72

5Ï€  = 5πθ/72

θ = 72°

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

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 × 2Ï€r cm

                     = Î¸/360 × 2Ï€(5)

3.5 = Î¸/360 × 2Ï€(5)

3.5 = 10π × θ/360

θ = 360 x 3.5/ (10π)

θ = 126/ π

Area of the sector = θ/360 × πr2

                             = (126/ Ï€)/ 360 × 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 72° at the centre. Find the length of the arc and area of the sector. 

Solution:

Radius = 35 cm

Angle subtended at the centre = 72°

Length of arc = θ/360 × 2πr cm

                      = 72/360 × 2Ï€(35)

                      = 14Ï€ 

                      = 14(22/7) 

                      = 44 cm

Area of the sector = θ/360 × πr2

                              = 72/360 × Ï€ 352

                              = (0.2) x (22/7) x 35 × 35

                             = 0.2 × 22 × 5 × 35

Area of the sector = (35 × 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 = θ/360 × 2πr m

Perimeter = l + 2r

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

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

27.2 – 11.4 = θ/360 × 2π x 5.7

15.8 = θ/360 × 2π x 5.7

θ = 158.8°

Area of the sector = θ/360 × πr2

Area of the sector = 158.8/360 × 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 × 2Ï€r cm

θ/360 × 2Ï€r cm + OA + OB = 27.2 m

θ/360 × 2Ï€r cm + 5.6 + 5.6 = 27.2 m

θ = 163.64°

Area of the sector = θ/360 × πr2

Area of the sector = 163.64/360 × Ï€ 5.62 

                             = 44.8

Therefore, the area of the sector = 44.8 m2



Last Updated : 03 May, 2021
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