In our everyday lives, we come across many plane figures, which are combinations of two or more plane figures. This exercise contains problems on finding the area of such combinations of plane figures. The RD Sharma Solutions Class 10, prepared by experts at BYJU’S, can be utilised by students to access detailed solutions to all exercise problems. Also, students can access the **RD Sharma Solutions for Class 10 Maths Chapter 15 Areas Related to Circles Exercise 15.4 **PDF provided below.

## RD Sharma Solutions for Class 10 Maths Chapter 15 Areas Related to Circles Exercise 15.4 Download the PDF

### Access RD Sharma Solutions for Class 10 Maths Chapter 15 Areas Related to Circles Exercise 15.4

**1. A plot is in the form of a rectangle ABCD having a semi-circle on BC, as shown in Fig.15.64. If AB = 60m and BC = 28m, find the area of the plot.**

**Solution:**

Given, ABCD is a rectangle.

So, AB = CD = 60 m

And, BC = AD = 28 m

For the radius of the semi-circle = BC/2 = 28/2 = 14 m

Now,

Area of the plot = Area of rectangle ABCD + Area of semi-circle

= (l x b) + ½ πr^{2}

= (60 x 28) + ½ (22/7)(14)^{2}

= 1680 + 308

= 1988 cm^{2}

**2.** **A playground has the shape of a rectangle, with two semi-circles on its smaller sides as diameters added to its outside. If the sides of the rectangle are 36m and 24.5m, find the area of the playground. **

**Solution: **

Given,

Length of rectangle = 36 m

Breadth of rectangle = 24.5 m

Radius of the semi-circle = Breath/2 = 24.5/2 = 12.25 m

Now,

Area of the playground = Area of the rectangle + 2 x Area of semi-circles

= l x b + 2 x ½ (πr^{2})

= (36 x 24.5) + (22/7) x 12.25^{2}

= 882 + 471.625 = 1353.625

Thus, the area of the playground is 1353.625 m^{2}

**3. Find the area of the circle in which a square of area 64 cm ^{2} is inscribed. **

**Solution:**

Given,

The area of the square inscribed the circle = 64 cm^{2}

Side^{2} = 64

Side = 8 cm

So, AB = BC = CD = DA = 8 cm

In triangle ABC, by Pythagoras’ theorem, we have

AC^{2} = AB^{2} + BC^{2}

AC^{2} = 8^{2} + 8^{2}

AC^{2} = 64 + 64 = 128

AC = √128 = 8√2 cm

Now, as angle B = 90^{o}, and AC is the diameter of the circle.

The radius is AC/2 = 8√2/2 = 4√2 cm

Thus, the area of the circle = πr^{2} = 3.14(4√2)^{2}

= 100.48 cm^{2}

**4. A rectangular piece is 20m long and 15m wide. From its four corners, quadrants of radii 3.5m have been cut. Find the area of the remaining part.**

**Solution: **

Given,

The length of the rectangle = 20 m

The breadth of the rectangle = 15 m

The radius of the quadrant = 3.5 m

So,

Area of the remaining part = Area of the rectangle – 4 x Area of one quadrant

= (l x b) – 4 x (¼ x πr^{2})

= (l x b) – πr^{2}

= (20 x 15) – (22/7)(3.5)^{2}

= 300 – 38.5

= 261.5 m^{2}

**5. In fig. 15.73, PQRS is a square of side 4 cm. Find the area of the shaded square. **

**Solution: **

We know that each quadrant is a sector of 90^{o} in a circle of 1 cm radius. In other words, it’s 1/4^{th} of a circle.

So, its area = ¼ x πr^{2}

= ¼ x (22/7)(1)^{2} = 22/28 cm^{2}

And, the area of the square = side^{2} [Given, side = 4 cm]

= 4^{2} = 16 cm^{2}

Area of the circle = πr^{2} = π(1)^{2} = 22/7 cm^{2} [Given, diameter = 2 cm, so radius = 1cm]

Thus,

The area of the shaded region = Area of the square – Area of the circle – 4 x Area of a quadrant

= 16 – 22/7 – (4 x 22/28)

= 16 – 22/7 – 22/7 = 16 – 44/7

= 68/7 cm^{2}

**6. Four cows are tethered at four corners of a square plot of side 50m, so that they just cannot reach one another. What area will be left un-grazed?**

**Solution: **

Given,

The ide of square plot = 50 m

The radius of a quadrant = 25 m

So, we can tell

Area of plot left un-grazed = Area of the plot – 4 x (Area of a quadrant)

= Side^{2} – 4 x (¼ x πr^{2})

= 50^{2 }– 22/7 x (25)^{2}

= 2500 – 1964.28

= 535.72 m^{2}

**7. A cow is tied with a rope of length 14 m at the corner of a rectangle field of dimensions 20 m x 16 m, find the area of the field in which the cow can graze. **

**Solution: **

The dotted portion indicates the area over which the cow can graze.

It’s clearly seen that the shaded area is the area of a quadrant of a circle of radius equal to the length of the rope.

Thus, the required area = ¼ x πr^{2}

= ¼ x 22/7 x 14 x 14

= 154

Hence, the area of the field in which the cow can graze is 154 cm^{2}

**8. A calf is tied with a rope of length 6 m at the corner of a square grassy lawn of side 20 m. If the length of the rope is increased by 5.5 m, find the increase in the area of the grassy lawn in which the calf can graze. **

**Solution: **

Given,

The initial length of the rope = 6 m

Then the rope is said to be increased by 5.5m.

So, the increased length of the rope = (6 + 5.5) = 11.5 m

We know that the corner of the lawn is a quadrant of a circle.

Thus,

The required area = ¼ x π(11.5)^{2} – ¼ x π(5.5)^{2}

= ¼ x 22/7 (11.5^{2 }– 6^{2})

= ¼ x 22/7 (132.25 – 36)

= ¼ x 22/7 x 96.25

= 75.625 cm^{2}

**9. A square tank has its side equal to 40 m. There are four semi-circular grassy plots all around it. Find the cost of turfing the plot at Rs 1.25 per square meter. **

**Solution: **

Given,

Side of the square tank = 40 m

And, the diameter of the semi-circular grassy plot = side of the square tank = 40 m

Radius of the grassy plot = 40/2 = 20 m

Then,

The area of the four semi-circular grassy plots = 4 x ½ πr^{2}

= 4 x ½ (3.14)(20)^{2}

= 2512 m^{2}

Rate of turfing the plot = Rs 1.25 per m^{2}

So, the cost for 2512 m^{2} = (1.25 x 2512) = Rs 3140

**10. A rectangular park is 100 m by 50 m. It is surrounded by semi-circular flower beds all round. Find the cost of levelling the semi-circular flower beds at 60 paise per square metre.**

**Solution: **

Given,

Length of the park = 100 m and the breadth of the park = 50 m

The radius of the semi-circular flower beds = Half of the corresponding side of the rectangular park

The radius of the bigger flower bed = 100/2 = 50 m

And the radius of the smaller flower bed = 50/2 = 25 m

Total area of the flower beds = 2[Area of bigger flower bed + Area of smaller flower bed]

= 2[½ π(50)^{2} + ½ π(25)^{2}]

= π[(50)^{2} + (25)^{2}]

= 3.14 x [2500 + 625]

= 9812.5 m^{2}

Now, the rate of levelling flower bed = 60 paise per m^{2}

Therefore,

The total cost of levelling = 9812.5 x 60 = 588750 paise

= Rs 5887.50

**11. The inner perimeter of a running track (as shown in Fig.) is 400 m. The length of each of the straight portions is 90 m, and the ends are semi-circles. If the track is everywhere 14 m wide, find the area of the track. Also, find the length of the outer running track. **

**Solution: **

Let the radius of the inner semi-circle = r

And that of the outer semi-circle = R

Given,

Length of the straight portion = 90 m

Width of the track = 14 m

The inner perimeter of the track = 400 m

But,

Inner perimeter of the track = BF + FRG + GC + CQB = 400

90 + πr + 90 + πr = 400

2 πr + 180 = 400

2 x 22/7 x r = 220

r = 35 m

So, the radius of the outer semi-circle = 35 + 14 = 49 m

Now,

Area of the track = 2[Area of the rectangle AEFB + Area of semi-circle APD – Area of semi-circle BQC]

= 2[90 x 14 + ½ x 22/7 x 49^{2} – ½ x 22/7 x 35^{2}]

= 2[1260 + 11 x 7 x 49 – 11 x 5 x 35]

= 2 [1260 + 3773 – 1925]

= 3 x 3108

= 6216 m^{2}

Thus,

The length of the outer running track = AE + APD + DH + HSE

= 90 + πR + 90 + πR

= 180 + 2 πR

= 180 + 2 x 22/7 x 49

= 180 + 308

= 488 m

**12. Find the area of Fig. 15.76 in square cm, correct to one place of decimal. **

**Solution: **

The radius of the semi-circle = 10/2 = 5 cm

It’s seen that,

The area of the figure = Area of square + Area of semi-circle – Area of triangle AED

= 10 x 10 + ½ πr^{2} – ½ x 6 x 8

= 100 + ½ (22/7)(5)^{2} – 24

= (700 + 275 – 168)/7

= (807)/7

= 115.3 cm^{2}

**13. From a rectangular region ABCD with AB = 20 cm, a right angle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as the diameter, a semicircle is added outside the region. Find the area of the shaded region. (π = 22/7) **

**Solution: **

Given,

Length of the rectangle ABCD = 20 cm

AE = 9 cm and DE = 12 cm

The radius of the semi-circle = BC/ 2 or AD/2

Now, using Pythagoras’ theorem in triangle AED,

AD = √(AE^{2} + ED^{2}) = √(9^{2} + 12^{2})

= √(81 + 144)

= √(225) = 15 cm

So, the area of the rectangle = 20 x 15 = 300 cm^{2}

And, the area of the triangle AED = ½ x 12 x 9 = 54 cm^{2}

The radius of the semi-circle = 15/2 = 7.5 cm

Area of semi-circle = ½ π(15/2)^{2} = ½ x 3.14 x 7.5^{2} = 88.31 cm^{2}

Thus,

The area of the shaded region = Area of the rectangle ABCD + Area of semi-circle – Area of triangle AED

= 300 + 88.31 – 54

= 334.31 cm^{2}

**14. From each of the two opposite corners of a square of side 8 cm, a quadrant of a circle of radius 1.4 cm is cut. Another circle of radius 4.2 cm is also cut from the centre, as shown in Fig. Find the area of the remaining (shaded) portion of the square. (Use π = 22/7)**

**Solution: **

Given,

The side of the square = 8 cm

The radius of the circle = 4.2 cm

The radius of the quadrant = 1.4 cm

Thus,

Area of the shaded portion = Area of square – Area of circle – 2 x Area of the quadrant

= side^{2} – πr^{2 }– 2 x ½ πr^{2}

= 8^{2} – π(4.2)^{2 }– 2 x ½ π(1.4)^{2}

= 64 – 22/7(4.2 x 4.2) – 22/7(1.4 x 1.4)

= 64 – 388.08/7 – 21.56/7

= 5.48 cm^{2}

**15. ABCD is a rectangle with AB = 14 cm and BC = 7 cm. Taking DC, BC and AD as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region.**

**Solution: **

Given,

ABCD is a rectangle with AB = 14 cm and BC = 7 cm

It’s seen that,

The area of shaded region = Area of rectangle ABCD + 2 x Area of a semi-circle with AD and BC as diameters – Area of the semi-circle with DC as the diameter

= 14 x 7 + 2 x ½ π(7/2)^{2 }– ½ π(7)^{2 }

= 98 + 22/7 x (7/2)^{2 }– ½ (22/7)(7)^{2 }

= 98 + 77/2 – 77

= 59.5 cm^{2}

**16. ABCD is rectangle, having AB = 20 cm and BC = 14 cm. Two sectors of 180° have been cut off. Calculate:**

**(i) the area of the shaded region. **

**(ii) the length of the boundary of the shaded region.**

**Solution: **

Given,

Length of the rectangle = AB = 20 cm

Breadth of the rectangle = BC = 14 cm

(i) Area of the shaded region = Area of rectangle – 2 x Area of the semi-circle

= l x b – 2 x ½ πr^{2}

= 20 x 14 – (22/7) x 7^{2}

= 280 – 154

= l26 cm^{2}

(ii) Length of the boundary of the shaded region = 2 x AB + 2 x Circumference of a semi-circle

= 2 x 20 + 2 x πr

= 40 + 2 x (22/7) x 7

= 40 + 44

= 84 cm

**17. The square ABCD is divided into five equal parts, all having the same area. The central part is circular, and the lines AE, GC, BF and HD lie along the diagonals AC and BD of the square. If AB = 22 cm, find:**

**(i) the circumference of the central part. (ii) the perimeter of part ABEF.**

**Solution: **

Given,

Side of the square = 22 cm = AB

Let the radius of the centre part be r cm.

Then, the area of the circle = 1/5 x the area of the square

πr^{2 }= 1/5 x 22^{2}

22/7 x r^{2} = (22 x 22)/ 5

r = 154/5 = 5.55 cm

(i) Circumference of central part = 2πr = 2(22/7)(5.55) = 34.88 cm

(ii) Let O be the centre of the central part. Then, it’s clear that O is also the centre of the square as well.

Now, in triangle ABC,

By Pythagoras theorem,

AC^{2} = AB^{2} + BC^{2} = 22^{2} + 22^{2} = 2 x 22^{2}

AC = 22√2

Since diagonals of a square bisect each other,

AO = ½ AC = ½ (22√2) = 11√2 cm

And,

AE = BF = OA – OE = 11√2 – 5.55 = 15.51 – 5.55 = 9.96 cm

EF = ¼(Circumference of the circle) = 2πr/4

= ½ x 22/7 x 5.55 = 8.72 cm

Thus, the perimeter of the part ABEF = AB + AE + EF + BF

= 22 + 2 x 9.96 + 8.72

= 50.64 cm

**18. In the figure, find the area of the shaded region. (Use π = 3.14)**

**Solution: **

The side of the square = 14 cm

So, it’s area = 14^{2} = 196 cm^{2}

Let’s assume the radius of each semi-circle is r cm.

Then,

r + 2r + r = 14 – 3 – 3

4r = 8

r = 2

The radius of each semi-circle is 2 cm.

The area of 4 semi-circles = (4 x ½ x 3.14 x 2 x 2) = 25.12 cm^{2}

Now, length of side of the smaller square = 2r = 2 x 2 = 4 cm

Thus, the area of smaller square = 4×4 = 16 cm^{2}

Area of unshaded region = Area of 4 semi-circles + Area of smaller square

= (25.12 + 16) = 41.12 cm^{2}

Therefore,

The area of shaded region = Area of the square ABCD – Area of the unshaded region

= (196 – 41.12) = 154.88 cm^{2}

**19. OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the (i) quadrant OACB (ii) shaded region.**

**Solution: **

Given,

The radius of the small quadrant, r = 2 cm

The radius of the big quadrant, R = 3.5 cm

(i) Area of quadrant OACB = ¼ πR^{2}

= ¼ (22/7)(3.5)^{2}

= 269.5/28 = 9.625 cm^{2}

(ii) Area of the shaded region = Area of the big quadrant – Area of the small quadrant

= ¼ π(R^{2} – r^{2})

= ¼ (22/7)(3.5^{2} – 2^{2})

= ¼ (22/7)(12.25 – 4)

= ¼ (22/7)(8.25)

= 6.482 cm^{2}

**20. A square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 21 cm, find the area of the shaded region.**

**Solution: **

Given,

Side of the square = 21 cm = OA

Area of the square = OA^{2} = 21^{2} = 441 cm^{2}

Diagonal of the square OB = √2 OA = 21√2 cm

And, from the fig., it’s seen that

The diagonal of the square is equal to the radius of the circle, r = 21√2 cm

So, the area of the quadrant = ¼ πr^{2} = ¼ (22/7)(21√2)^{2} = 693 cm^{2}

Thus,

The area of the shaded region = Area of the quadrant – Area of the square

= 693 – 441

= 252 cm^{2}

**21. OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.**

**Solution: **

Given,

OABC is a square of side 7 cm

So, OA = AB = BC = OC = 7 cm

Area of square OABC = side^{2} = 7^{2} = 49 cm^{2}

And given, OAPC is a quadrant of a circle with centre O.

So, the radius of the quadrant = OA = OC = 7 cm

Area of the quadrant OAPC = 90/360 x πr^{2}

= ¼ x (22/7) x 7^{2}

= 77/2 = 38.5 cm^{2}

Thus,

Area of shaded portion = Area of square OABC – Area of quadrant OAPC

= (49 – 38.5) = 10.5 cm^{2}

**22. OE = 20 cm. In sector OSFT, square OEFG is inscribed. Find the area of the shaded region.**

**Solution: **

It’s seen that OEFG is a square of side 20 cm.

So its diagonal = √2 side = 20√2 cm

And, the radius of the quadrant = Diagonal of the square

Radius of the quadrant = 20√2 cm

So,

Area of the shaded portion = Area of the quadrant – Area of the square

= ¼ πr^{2} – side^{2}

= ¼ (22/7)(20√2)^{2} – (20)^{2}

= ¼ (22/7)(800) – 400

= 400 x 4/7 = 1600/7 = 228.5 cm^{2}

**23. Find the area of the shaded region in Fig., if AC = 24 cm, BC = 10 cm, and O is the centre of the circle.**

**Solution: **

Given,

AC = 24 cm and BC = 10 cm

Since AB is the diameter of the circle,

Angle ACB = 90^{o}

So, using Pythagoras’ theorem,

AB^{2} = AC^{2} + BC^{2} = 24^{2} + 10^{2} = 576 + 100 = 676

AB = √676 = 26 cm

Thus, the radius of the circle = 26/2 = 13 cm

The area of shaded region = Area of semi-circle – Area of triangle ACB

= ½ πr^{2} – ½ x b x h

= ½ (22/7)13^{2} – ½ x 10 x 24

= 265.33 – 120

= 145.33 cm^{2}

**24. A circle is inscribed in an equilateral triangle ABC of side 12 cm, touching its sides (fig.,). Find the radius of the inscribed circle and the area of the shaded part.**

**Solution:**

Given,

An equilateral triangle of sides = 12 cm

Area of the equilateral triangle = √3/4(side)^{2}

= √3/4(12)^{2} = 36√3 cm^{2}

Perimeter of triangle ABC = 3 x 12 = 36 cm

So,

The radius of incircle = Area of triangle/ ½ (perimeter of triangle)

= 36√3/ ½ x 36

= 2√3 cm

Therefore,

Area of the shaded part = Area of the equilateral triangle – Area of the circle

= 36√3 – πr^{2}

= 36(1.732) – (3.14)(2√3)^{2}

= 62.352 – 37.68

= 24.672 cm^{2}

**25. In fig., an equilateral triangle ABC of side 6 cm has been inscribed in a circle. Find the area of the shaded region. (Take π = 3.14)**

**Solution: **

Given,

Side of the equilateral triangle = 6 cm

And,

The area of the equilateral triangle = √3/4(side)^{2}

= √3/4(6)^{2}

= √3/4(36)

= 9√3 cm^{2}

Let us mark the centre of the circle as O, OA and OB are the radii of the circle.

In triangle BOD,

sin 60^{o} = BD/ OB

√3/2 = 3/ OB

OB = 2√3 cm = r

Therefore,

The area of the shaded region = Area of the circle – Area of the equilateral triangle

= πr^{2 }– 9√3

= 3.14 x (2√3)^{2} – 9√3

= 3.14 x 12 – 9 x 1.732

= 37.68 – 15.588

= 22.092 cm^{2}

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