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

Question 1. Find, in terms of π, the length of the arc that subtends an angle of 30^o 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 × 2πr 

Length of arc = 30/360 × 2π × 4 cm 

                      = 2π/3 

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

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

Solution:

Length of arc = 5π/3 cm

Length of arc = θ/360 × 2πr cm

5π/3 cm = θ/360 × 2πr cm

θ = 60°

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

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

Solution:

Length of arc = 20π cm

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

Length of arc = θ/360 × 2πr cm

θ/360 × 2πr cm = 144/360 × 2πr cm = 4π/5 × r cm

20π cm = 4π/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 π, the radius of the circle. 

Solution:

Length of arc = 15 cm

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

Length of arc = θ/360 × 2πr cm

                = 45/360 × 2πr cm

15 cm = 45/360 × 2π × r cm

15 = πr/4

Radius = 15×4/ π = 60/π

Therefore, the radius of the circle is 60/π cm.

Question 5. Find the angle subtended at the centre of a circle of radius ‘a’ cm by an arc of length (aπ/4) cm. 

Solution:

Radius = a cm

Length of arc = aπ/4 cm

θ = angle subtended at the centre of circle

Length of arc = θ/360 × 2πr cm

θ/360 × 2πa cm = aπ/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 × π4²

                             = 1/12 × π16 

                             = 4π/3 cm² 

                             = 4.19 cm² 

Therefore, the area of the sector of the circle = 4.19 cm² 

Question 7. A sector of a circle of radius 8 cm contains an angle of 135^o. 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

                             = 24π 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 π 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 × π2²

                            = πθ/90

π = π 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 5π cm². Find the angle contained by the sector. 

Solution:

Radius = 5 cm

Area of sector of circle = 5π cm²

Area of the sector = θ/360 × πr²

                             = θ/360 × π5²

                             = 5πθ/72

5π = 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 = θ/360 × 2πr cm

                     = θ/360 × 2π(5)

3.5 = θ/360 × 2π(5)

3.5 = 10π × θ/360

θ = 360 x 3.5/ (10π)

θ = 126/ π

Area of the sector = θ/360 × πr²

                             = (126/ π)/ 360 × π(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 × 2π(35)

                      = 14π 

                      = 14(22/7) 

                      = 44 cm

Area of the sector = θ/360 × πr²

                              = 72/360 × π 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 = θ/360 × 2πr + OA + OB

27.2 = θ/360 × 2π 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 × π 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 = θ/360 × 2πr cm

θ/360 × 2πr cm + OA + OB = 27.2 m

θ/360 × 2πr 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 × π 5.6²

                             = 44.8

Therefore, the area of the sector = 44.8 m²