The problem of finding perimeters and areas related to circular figures is of great practical importance. This exercise contains problems on finding the perimeter and area of a circle as well as the area enclosed by two concentric circles. To access detailed solutions to problems, the RD Sharma Solutions Class 10 is available for students for the purpose of clearing doubts. Downloadable format of the RD Sharma Solutions for Class 10 Maths Chapter 15 Areas Related To Circles Exercise 15.1 PDF is provided below.

## RD Sharma Solutions for Class 10 Chapter 15 Areas Related To Circles Exercise 15.1 Download PDF

### Access RD Sharma Solutions for Class 10 Chapter 15 Areas Related To Circles Exercise 15.1

**1. Find the circumference and area of a circle of radius ofÂ 4.2 cm.Â **

**Solution:**

Given,

Radius (r) = 4.2 cm

We know that,

Circumference of a circle = 2Ï€r

= 2 Ã— (22/7) Ã— 4.2 = 26.4 cm^{2}

Area of a circle = Ï€r^{2}

= (22/7) x 4.2^{2}

= 22 x 0.6 x 4.2 = 55.44 cm^{2}

Hence, the circumference and area of the circle is 26.4 cm^{2}Â and 55.44 cm^{2}.

Â

**2. Find the circumference of a circle whose area is 301.84 cm ^{2}.Â **

**Solution:**

Given,

Area of the circle = 301.84 cm^{2Â Â Â Â Â Â Â Â Â Â Â Â }

We know that,

Area of a Circle = Ï€r^{2 }= 301.84 cm^{2Â Â }

(22/7) Ã— r^{2}Â = 301.84

r^{2} = 13.72 x 7 = 96.04

r = âˆš96.04 = 9.8

So, the radius is = 9.8 cm.

Now, Circumference of a circle = 2Ï€r

= 2 Ã— (22/7) Ã— 9.8 = 61.6 cm

Hence, the circumference of the circle is 61.6 cm.

Â

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

**Solution:**

Given,

Circumference =Â 44 cm

We know that,

Circumference of a circle = 2Ï€r = 44 cm

2 Ã— (22/7) Ã— r = 44

r = 7 cm

Now, Area of a Circle = Ï€r^{2}

= (22/7) Ã— 7 Ã— 7

= 154 cm^{2}Â

Hence, area of the Circle = 154 cm^{2}Â

Â

**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

So, the diameter (d)Â = 2r [As radius is half the diameter]

We know that,

Circumference of a circle (C)Â = 2Ï€r

From the question,

Circumference of the circle exceeds its diameter by 16.8 cm

C = d + 16.8

2Ï€r = 2r + 16.8Â Â Â [d = 2r]

2Ï€r – 2r = 16.8

2r (Ï€ – 1) = 16.8

2r (3.14 – 1) = 16.8

r = 3.92 cm

Thus, radius = 3.92 cm

Now, the circumference of the circle (C) = 2Ï€r

C = 2 Ã— 3.14 Ã— 3.92

= 24.64 cm

Hence, the circumference of the circle is 24.64 cm.Â

Â

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

**Solution:**

Given,

Length of the string (l) = 28 m

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

We know that,

Area of a Circle = Ï€r^{2}

= (22/7) Ã— 28 Ã— 28 = 2464 m^{2}Â

Hence, the area of the circle which is same as the area the horse can graze is 2464 m^{2}Â

Â

**6. A steel wire when bent in the form of a square encloses an area of 121 cm ^{2}. If the same wire is bent in the form of a circle, find the area of the circle.Â **

**Solution:**

Given,

Area of the square = a^{2} = 121 cm^{2}Â

We know that,

Area of the circle = Ï€r^{2}

Area of a square = a^{2}

121 cm^{2}Â = a^{2}Â

So, a = 11 cmÂ

Thus, each side of the square = 11 cm

Now, the perimeter of the square =Â 4aÂ

= 4 Ã— 11 = 44 cmÂ

From the question, its understood that

Perimeter of the square = Circumference of the circleÂ

We know that, circumference of a circle (C)Â = 2Ï€r

4a = 2Ï€r

44 = 2(22/7)r

r = 7 cmÂ

Now, area of the Circle = Ï€r^{2}

= (22/7) Ã— 7 Ã— 7 = 154 cm^{2}

Hence, the area of the circle is 154 cm^{2}.

Â

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

**Solution:**

Letâ€™s consider the radius of two circles C_{1}Â and C_{2}Â be r_{1}Â and r_{2}.

We know that, Circumference of a circle (C)Â = 2Ï€r

And their circumference will be 2Ï€r_{1}Â and 2Ï€r_{2}.

So, their ratio is = r_{1}:Â r_{2}

Given, circumference of two circles is in a ratio of 2: 3

r_{1}: r_{2}Â =Â 2: 3

Then, the ratios of their areas is given as

= 4/9

Hence, ratio of their areas = 4: 9.Â

Â

**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.Â **

**Solution:**

Let the radii of the two circles be r_{1}Â and r_{2}.

And, the circumferences of the two circles be C_{1}Â and C_{2}.

We know that, circumference of circle (C) = 2Ï€r

Given,

Sum of radii of two circle i.e., r_{1}Â + r_{2}Â = 140 cmÂ …Â (i)Â

Difference of their circumference,

C_{1}Â â€“ C_{2}Â = 88 cm

2Ï€r_{1}Â – 2Ï€r_{2Â }= 88 cm

2(22/7)(r_{1}Â â€“ r_{2}) = 88 cm

(r_{1}Â â€“ r_{2)}Â = 14 cm

r_{1 =}Â r_{2}Â + 14…..**Â **(ii)**Â **

Substituting the value of r_{1Â }in equationÂ (i)**,Â **we have,

r_{2Â }+ r_{2}Â + 14 = 140

2r_{2}Â = 140 – 14

2r_{2Â }= 126

r_{2}Â = 63 cm

Substituting the value of r_{2Â }in equationÂ (ii)**,Â **we have,

r_{1Â }= 63 + 14Â = 77 cm

Therefore,

Diameter of circle 1 = 2r_{1Â }= 2 x 77 = 154 cm

Diameter of circle 2 = 2r_{2Â }= 2 Ã— 63 = 126 cm

**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: **

Given,

Radius of circle 1 = r_{1} = 15 cm

Radius of circle 2 = r_{2} = 18 cm

We know that, the circumference of a circle (C) = 2Ï€r

So, C_{1} = 2Ï€r_{1 }and C_{2} = 2Ï€r_{2}

Let the radius be r of the circle which is to be found and its circumference (C)

Now, from the question

C = C_{1} + C_{2}

2Ï€r = 2Ï€r_{1} + 2Ï€r_{2}

r = r_{1} + r_{2 }[After dividing by 2Ï€ both sides]

r = 15 + 18

r = 33 cm

Thus, the radius of the circle = 33 cm

**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:**

Given,

Radii of the two circles are 6 cm and 8 cm

Area of circle with radius 8 cm = Ï€ (8)^{2}Â = 64Ï€Â cm^{2}Â

Area of circle with radius 6cm = Ï€ (6)^{2}Â = 36Ï€Â cm^{2}Â

Sum of areas = 64Ï€Â + 36Ï€Â = 100Ï€Â cm^{2}Â

Let the radius of the circle be x cm

Area of the circle = 100Ï€Â cm^{2}Â (from above)

Ï€x^{2Â }= 100Ï€

x = âˆš100 = 10 cm

Therefore, the radius of the circle is 10 cm.

Â

**11. The radii of two circles are 19 cm and 9 cm respectively. Find the radius and area of the circle which has circumferences is equal to sum of the circumference of two circles.Â **

**Solution:**

Given,

Radius of circle 1 = r_{1} = 19 cm

Radius of circle 2 = r_{2} = 9 cm

We know that, the circumference of a circle (C) = 2Ï€r

So, C_{1} = 2Ï€r_{1 }and C_{2} = 2Ï€r_{2}

Let the radius be r of the circle which is to be found and its circumference (C)

Now, from the question

C = C_{1} + C_{2}

2Ï€r = 2Ï€r_{1} + 2Ï€r_{2}

r = r_{1} + r_{2 }[After dividing by 2Ï€ both sides]

r = 19 + 9

r = 28 cm

Thus, the radius of the circle = 28 cm

So, the area of required circle = Ï€r^{2}Â = (22/7) Ã— 28 Ã— 28 = 2464 cm^{2}Â

Â

**12. The area of a circular playground is 22176 m ^{2}. Find the cost of fencing this ground at the rate of â‚¹50 per metre. **

**Solution: **

Given,

Area of the circular playground = 22176 m^{2}

And the cost of fencing per metre = â‚¹50

If the radius of the ground is taken as r.

Then, its area = Ï€r^{2}

Ï€r^{2} = 22176

r^{2} = 22176(7/22) = 7056

Taking square root on both sides, we have

r = 84 m

We know that, fencing is done only on the circumference of the ground

Circumference of the ground = 2Ï€r = 2(22/7)84 = 528 m

So, the cost of fencing 528 m = â‚¹50 x 528 = â‚¹26400

Therefore, the cost of fencing the ground is â‚¹26400.

**13. The side of a square is 10 cm. Find the area of the circumscribed and inscribed circles.Â **

**Solution:**

For circumscribed circle:

Radius = diagonal of square/ 2

Diagonal of the square = side x âˆš2

= 10âˆš2 cm

Radius = (10 Ã— 1.414)/ 2 = 7.07 cm

Thus, the radius of the circumcircle = 7.07 cm

Then, its area is = Ï€r^{2} = (22/7) Ã— 7.07 Ã— 7.07 = 157.41 cm^{2Â }

Therefore, the Area of the Circumscribed circle is 157.41 cm^{2Â }

For inscribed circle:

RadiusÂ = side of square/ 2

= 10/ 2 = 5 m

Then, its area is = Ï€r^{2} =Â 3.14 Ã— 5 Ã— 5 = 78.5 cm^{2Â }

Thus,Â the area of the circumscribed circle is 157.41 cm^{2Â }and the area of the inscribed circle is 78.5 cm^{2}.

Â

**14. If a square is inscribed in a circle, find the ratio of areas of the circle and the square.Â **

**Solution:**

Let side of square be x cm which is inscribed in a circle.

Given,

Radius of circle (r) =Â 1/2Â (diagonal of square)

=Â 1/2(xâˆš2)Â

r =Â x/âˆš2Â

We know that, area of the square = x^{2}Â

And, the area of the circle = Ï€r^{2}

Therefore, the ratio of areas of the circle and the square = Ï€ : 2

Â

**15. The area of circle inscribed in an equilateral triangle is 154 cm ^{2}. Find the perimeter of the triangle.**

**Solution:**

Let the circle inscribed in the equilateral triangle be with a centre O and radius r.

We know that, Area of a Circle = Ï€r^{2}

But, the given that area is 154 cm^{2}.

(22/7) Ã— r^{2}Â = 154

r^{2} = (154 x 7)/22 = 7 Ã— 7 = 49

r = 7 cm

From the figure seen above, we infer that

At point M, BC side is tangent and also at point M, BM is perpendicular to OM.

We know that,

In an equilateral triangle, the perpendicular from vertex divides the side into two halves.

BM = Â½ x BC

Consider the side of the equilateral triangle be x cm.

After solving the above equation we get,

Therefore, the perimeter of the triangle is found to beÂ 42âˆš3 cm = 42(1.73) = 72.7 cm