Class 10 RD Sharma Solutions – Chapter 15 Areas Related to Circles – Exercise 15.1 | Set 1

Question 1. Find the circumference and area of a circle of radius of 4.2 cm. 

Solution:

Radius = 4.2 cm

Circumference of a circle = 2πr

                                       = 2 × (22/7) × 4.2 

                                       = 26.4 cm

Area of a circle = πr²

                        = (22/7) x 4.2²

                        = (22/7) × 4.2 × 4.2 

                        = 55.44 cm²

Therefore, circumference = 26.4 cm and area of the circle = 55.44 cm²

Question 2. Find the circumference of a circle whose area is 301.84 cm². 

Solution:

Area of circle = 301.84 cm²            

Area of a Circle = πr² = 301.84 cm²  

(22/7) × r² = 301.84

r² = 96.04

r = √96.04 = 9.8cm

Radius = 9.8 cm.

Circumference of a circle = 2πr

                                        = 2 × (22/7) × 9.8 

                                        = 61.6 cm

Therefore, the circumference of the circle = 61.6 cm.

Question 3. Find the area of a circle whose circumference is 44 cm.

Solution:

Circumference = 44 cm

2πr = 44 cm

2 × (22/7) × r = 44

r = 7 cm

Area of a Circle = πr²

                         = (22/7) × 7 × 7

                         = 154 cm²

Therefore, area of the Circle = 154 cm²

Question 4. The circumference of a circle exceeds the diameter by 16.8 cm. Find the circumference of the circle.

Solution:

Let the radius of the circle be r cm

Diameter (d) = 2r 

Circumference of a circle (C) = 2πr

C = d + 16.8

2πr = 2r + 16.8  

2πr – 2r = 16.8

2r (π – 1) = 16.8

2r (3.14 – 1) = 16.8

r = 3.92 cm

Radius = 3.92 cm

Circumference (C) = 2πr

C = 2 × 3.14 × 3.92

= 24.62 cm

Therefore, circumference of the circle = 24.64 cm.

Question 5. A horse is tied to a pole with 28 m long string. Find the area where the horse can graze.

Solution:

Length of the string = 28 m

Area the horse can graze is the area of the circle with a radius equal to the length of the string.

Area of a Circle = πr²

                         = (22/7) × 28 × 28 

                         = 2464 m²

Therefore, the area where horse can graze = 2464 m²

Question 6. A steel wire when bent in the form of a square encloses an area of 121 cm². If the same wire is bent in the form of a circle, find the area of the circle. 

Solution:

Area of the square = a²

                             = 121 cm²

Area of the circle = πr²

121 cm² = a²

Therefore, a = 11 cm

Perimeter of square = 4a

                                = 4 × 11 = 44 cm

Perimeter of the square = Circumference of the circle

Circumference = 2πr

44 = 2(22/7)r

r = 7 cm

Area of the Circle = πr²

                            = (22/7) × 7 × 7 

                           = 154 cm²

Therefore, the area of the circle = 154 cm².

Question 7. The circumference of two circles are in the ratio of 2:3. Find the ratio of their areas.

Solution:

Circumference of a circle (C) = 2πr

Circumference of first circle = 2πr₁

Circumference of second circle = 2πr₂.

2πr₁ : 2πr₂ = 2:3

Therefore,

r₁: r₂ = 2: 3

Area of circle 1 = (πr₁)²

Area of circle 2 = (πr₂)²

Ratio = 2²:3²

         = 4/9

Therefore, ratio of areas = 4: 9.

Question 8. The sum of the radii of two circles is 140 cm and the difference of their circumference is 88 cm. Find the diameters of the circles. 

Solution:

Sum of radii of two circles i.e., r₁ + r₂ = 140 cm … (i)

Difference of their circumference,

C₁ – C₂ = 88 cm

2πr₁ – 2πr₂ = 88 cm

2(22/7)(r₁ – r₂) = 88 cm

(r₁ – r₂₎ = 14 cm

r_{1 =} r₂ + 14….. (ii)

From (i) and (ii)

r₂ + r₂ + 14 = 140

2r₂ = 140 – 14

2r₂ = 126

r₂ = 63 cm

r₁ = 63 + 14 = 77 cm

Therefore,

Diameter of circle 1 = 2 x 77 = 154 cm

Diameter of circle 2 = 2 × 63 = 126 cm

Question 9. Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii 15cm and 18cm.

Solution:

Radius of circle 1 = r₁ = 15 cm

Radius of circle 2 = r₂ = 18 cm

C₁ = 2πr_{1 ,} C₂ = 2πr₂

C = C₁ + C₂

2πr = 2πr₁ + 2πr₂

r = r₁ + r₂ 

r = 15 + 18

r = 33 cm

Therefore, the radius of the circle = 33 cm

Question 10. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of two circles.

Solution:

Radii of the two circles are 6 cm and 8 cm

Area of circle with radius 8 cm = π (8)²

                                                  = 64π cm²

Area of circle with radius 6cm = π (6)²

                                                = 36π cm²

Sum of areas = 64π + 36π = 100π cm²

Let the radius of the circle be r cm

Area of the circle = 100π cm²

πr² = 100π

r= √100 = 10 cm

Therefore, the radius of the circle = 10 cm.

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