# RD SHARMA Solutions for Class 10 Maths Chapter 13 - Areas Related to Circles

## Chapter 13 - Areas Related to Circles Exercise Ex. 13.1

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

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 the two circles.

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

Area of a circle = πr^{2} = (22/7) × 28 × 28 = 2464 cm^{2}

The area of a circular playground is 22176 m^{2}. Find the cost of fencing this ground at the rate of Rs. 50 per metre.

A park is in the form of a rectangle 120 m x 100 m. At the centre of the park there is a circular lawn. The area of park excluding lawn is 8700 m^{2}. Find the radius of the circular lawn. (Use = 22/7).

An archery target has three regions formed by three concentric circles as shown in figure. If the diameters of the concentric circles are in the ratio 1 : 2 : 3, then find the ratio of the areas of three regions.

The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/hr?

A circular pond is 17.5 m in diameter. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of Rs.25 per m^{2}.

A circular park is surrounded by a road 21 m wide. If the radius of the park is 105 m, find the area of the road.

A square of diagonal 8 cm is inscribed in a circle. Find the area of the region lying inside the circle and outside the square.

Find the area enclosed between two concentric circles of radii 3.5 cm and 7 cm. A third concentric circle is drawn outside the 7 cm circle, such that the area enclosed between it and the 7 cm circle is same as that between the two inner circles. Find the radius of the third circle correct to one decimal place.

A path of width 3.5 m runs around a semi-circular grassy plot whose perimeter is 72 m. Find the area of the path. (Use π = 22/7)

A circular pond is of diameter 17.5 m. It is surrounded by a 2 m wide path. Find the cost of constructing the path at the rate of Rs. 25 per square meter (π = 3.14)

The outer circumference of a circular race-track is 528 m. The track is everywhere 14 m wide. Calculate the cost of levelling the track at the rate of 50 paise per square metre (Use = 22/7).

Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is h (2r + h).

## Chapter 13 - Areas Related to Circles Exercise Ex. 13.2

The area of a sector of a circle of radius 5 cm is 5 cm^{2}. Find the angle contained by the sector.

Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm.

The perimeter of a scetor of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.

The perimeter of a certain sector of a circle of radius 5.6 m is 27.2 m. Find the area of the sector.

A sector of 56^{o} cut out from a circle contains area 4.4 cm^{2}. Find the radius of the circle.

Area of a sector of central angle 200° of a circle is 770 cm^{2}. Find the length of the corresponding arc of this sector.

The length of minute hand of a clock is 5 cm. Find the area swept by the minute hand during the time period 6:05 am and 6:40 am.

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

*Answer does not match with textbook answer.

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find

(i) the length of arc

(ii) area of the sector formed by the arc. (use π = 22/7)

From a circular piece of cardboard of radius 3 cm two sectors of 90° have been cut off. Find the perimeter of the remaining portion nearest hundredth centimeters. (Take π = 22/7)

*Note: Answer given in the book is incorrect.

The area of a sector is one-twelfth that of the complete circle. Find the angle of the sector.

AB is a chord of a circle with centre O and radius 4 cm. AB is of length 4 cm. Find the area of the sector of the circle formed by chord AB.

## Chapter 13 - Areas Related to Circles Exercise Ex. 13.3

A chord PQ of length 12 cm subtends an angle of 120^{o} at the centre of a circle. Find the area of the minor segment cut off by the chord PQ.

A chord 10 cm long is drawn in a circle whose radius is cm. Find area of both the segments. (Take = 3.14).

Find the area of the minor segment of a circle of radius 14 cm, when the angle of the corresponding sector is 60°.

A chord of a circle of radius 10 cm subtends an angle of 90° at the centre. Find the area of the corresponding major segment of the circle. (Use π = 3.14)

Note: Question modified

The radius of a circle with centre O is 5 cm. Two radii OA and OB are drawn at right angles to each other. Find the areas of the segments made by the chord AB. (π = 3.14)

## Chapter 13 - Areas Related to Circles Exercise Ex. 13.4

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

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

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

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

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

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 area of the grassy lawn in which the calf can graze.

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 (Ise = 3.14).

The inside perimeter of a running track (shown in Fig.) is 400 m. The length of each of the straight portion 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.

Find the area of Fig., in square cm, correct to one place of decimal. (Take π = 22/7).

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 diameter, a semicircle is added on outside the region. Find the area of the shaded region. (π = 22/7)

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)

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.

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.

The square ABCD is divided into five equal parts, all having 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 the part ABEF.

In figure, find the area of the shaded region.

(Use π = 3.14)

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.

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

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

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.

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

*Answer is not matching with textbook.

Find the area of a shaded region in the given figure, where a circular arc of radius 7 cm has been drawn with vertex A of an equilateral triangle ABC of side 14 cm as centre.

A regular hexagon is inscribed in a circle. If the area of hexagon is , find the area of the circle. (Use π it = 3.14)

Consider the following figure:

ABCDEF is a regular hexagon with centre O
(Fig.,). If the area of triangle OAB is 9 cm^{2}, find the area of:
(i) the hexagon and (ii) the circle in which the hexagon is inscribed.

(i)

According to the figure in the question, there are 6 triangles.

Area of one triangle is 9 cm^{2}.

Area of hexagon = 6 × 9 = 54 cm^{2}

(ii)

Area
of the equilateral triangle = 9 cm^{2}

Area of the circle in which the hexagon is inscribed

=

=

=

=
65.26 cm^{2}

NOTE: Answer not matching with back answer.

Four equal circles, each of radius 5 cm, touch each other as shown in Fig. Find the area included between them (Take π = 3.14)

A child makes a poster on a chart paper drawing a square ABCD of side 14 cm. She draws four circles with centre A, B, C and D in which she suggests different ways to save energy. The circles are drawn in such a way that each circle touches externally two of the three remaining circles. In the shaded region she write a message 'Save Energy'. Find the perimeter and area of the shaded region. (Use π = 22/7)

The diameter of a coin is 1 cm. If four such coins be placed on a table so that the rim of each touches that of the other two, find the area of the shaded region (Take π = 3.1416)

Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions 14 cm × 7 cm. find the area of the remaining card board. (π = 22/7)

AB and CD are two diameters of a circle perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

PSR, RTQ and PAQ are three semi-circles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region.

Two circles with centres A and B touch each other at the point C. If AC = 8 cm and AB = 3 cm, find the area of the shaded region.

ABCD is a square of side 2a. Find the ratio between

(i) the circumferences

(ii) the areas of the incircle and the circum-circle of the square.

There are three semicircles, A, B and C having diameter 3 cm each, and another semicircle E having a circle D with diameter 4.5 cm are shown. Calculate:

(i) the area of the shaded region

(ii) the cost of painting the shaded region at the rate of 25 paise per cm^{2}, to the nearest rupee.

O is the centre of a circular arc and AOB is a straight line. Find the perimeter and the area of the shaded region correct to one decimal place. (Take π = 3.142)

The boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, find (i) the length of the boundary (ii) the area of the shaded region.

Ab = 36 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all the three circles. Find the area of the shaded region.

Shows a kite in which BCD is the shape of a quadrant of a circle of radius 42 cm. ABCD is a square and Δ CEF is an isosceles right angled triangle whose equal sides are 6 cm long. Find the area of the shaded region.

ABCD is a trapezium of area 24.5 cm^{2}. In it, AD ∥ BC, ∠DAB = 90°, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region.

(π = 22/7)

ABCD is a trapezium with AB ∥ DC, AB = 18 cm, DC = 32 cm and the distance between AB and DC is 14 cm. Circles of equal radii 7 cm with centres A, B, C and D have been drawn. Then, find the area of the shaded region of the figure. (π = 22/7)

Since the data given in the question seems incomplete and inconsistent with the figure, we make the following assumptions to solve it:

1. ABCD a symmetric trapezium with AD = BC

2. AD = BC = 14 cm (the distance between AB and CD is not 14 cm)

Draw perpendiculars to CD from A and B to divide the trapezium into one rectangle and two congruent right angled triangles.

The base of the right angled triangle=(CD - AB) ÷ 2

=(32 - 18) ÷ 2=7 cm

cos∠D = base ÷ hypotenuse = 7 ÷ 14 =1/2

m∠D = 60°

Hence, m∠A = 120°

*Answer is not matching with textbook answer.

Sides of a triangular field are 15 m, 16 m and 17 m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7 m each to graze in the field. Find the area of the field which cannot be grazed by three animals.

In the given Fig., the side of square is 28 cm, and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.

According to the question,

Side of a square is 28 cm.

Radius of a circle is 14 cm.

Required area = Area of the square + Area of the two circles - Area of two quadrants …(i)

Area of the square = 28^{2} = 784
cm^{2}

Area of the two circles = 2πr^{2}

=

= 1232 cm^{2}

Area of two quadrants =

=

= 308 cm^{2}

Required area = 784 + 1232 - 308 = 1708
cm^{2}

NOTE: Answer not matching with back answer.

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park?

According to the question,

For a cylindrical tank

d = 2 m, r = 1 m, h = 5 m

Volume of the tank = πr^{2}h

=

=

After recycling, this water is used irrigate a park of a hospital with length 25 m and breadth 20 m.

If the tank is filled completely, then

Volume of cuboidal park = Volume of tank

h = 0.0314 m = 3.14 cm = p cm

## Chapter 13 - Areas Related to Circles Exercise 13.69

If the circumference and the area of a circle are numerically equal, then diameter of the circle is

Correct Option :- (D)

If the difference between the circumference and radius of a circle is 37 cm., then using π = , the circumference (in cm) of the circle is

(a) 154

(b) 44

(c) 14

(d) 7

According to the question,

Circumference of a circle =

=

= 44 cm

A write can be bent in the form of a circle of radius 56 cm. If it is bent in the form of a square, then its area will be

(a) 3520 cm^{2}

(b) 6400 cm^{2}

(c) 7744 cm^{2}

(d) 8800 cm^{2}

Correct option (c)

correct option - (c)

A circular park has a path of uniform width around it. The difference between the outer and inner circumferences of the circular path is 132 m. Its width is

(a) 20 m

(b) 21 m

(c) 22 m

(d) 24 m

correct option - (b)

The radius of a wheel is 0.25 m. The number of revolutions it will make to travel a distance of 11 km will be

(a) 2800

(b) 4000

(c) 5500

(d) 7000

Correct Option: d

The ratio of the outer and inner perimeters of a circular path is 23:22. If the path is 5m wide, the diameter of the inner circle is

(a) 55m

(b) 110 m

(c) 220 m

(d) 230 m

Correct Option: (c)

Correct option - (c)

Correct option (c)

Correct Option ( d )

Correct option (a)

The perimeter of a triangle is 30 cm and the circumference of its incircle is 88 cm. The area of the triangle is

a. 70 cm^{2}

b. 140 cm^{2}

c. 210 cm^{2}

d. 420 cm^{2}

Let r be the radius of the circle.

2pr = 88

Perimeter of a triangle = 30 cm

Semi-perimeter = 15 cm

Hence,

Area of a triangle = r × s …(r = incircle radius, s =semi perimeter)

= 14 × 15

=
210 cm^{2}

Correct option - (c)

## Chapter 13 - Areas Related to Circles Exercise 13.70

If the circumference of a circle increases from 4π to 8π, then its area is

(a) halved

(b) doubled

(c) tripled

(d) quadrupled

If the radius of a circle is diminished by 10%, then its area is diminished by

(a) 10%

(b) 19%

(c) 20%

(d) 36%

If the perimeter of a semi-circular protractor is 36 cm, then its diameter is

(a) 10 cm

(b) 12 cm

(c) 14 cm

(d) 16 cm

If the perimeter of a sector of a circle of radius 6.5 cm is 29 cm, then its area is

(a) 58 cm^{2}

(b) 52 cm^{2}

(c) 25 cm^{2}

(d) 56 cm^{2}

If the area of a sector of a circle bounded by an arc of length 5π cm is equal to 20π cm^{2 }, then its radius is

(a) 12 cm

(b) 16 cm

(c) 8 cm

(d) 10 cm

The area of the circle that can be inscribed in a square of side 10 cm is

(a) 40 π cm^{2}

(b) 30 π cm^{2}

(c) 100 π cm^{2}

(d) 25 π cm^{2}

Correct option: (d)

Diameter of circle = side of square

2r = 10

r = 5 cm

Area of circle = πr^{2} = 25 π cm^{2}

If the difference between the circumference and radius of a circle is 37 cm, then its area is

(a) 154 cm^{2}

(b) 160 cm^{2}

(c) 200 cm^{2 }

(d) 150 cm^{2}

## Chapter 13 - Areas Related to Circles Exercise 13.71

The area of a circular path of uniform width h surrounding a circular region of radius r is

(a) π (2r + h) r

(b) π (2r + h) h

(c) π (h + r) r

(d) π (h + r) h

Correct option: (b)

Inner radius = r

outer radius = r + h

area of shaded region = area of outer circle - area of inner circle

= π (r + h)^{2} - πr^{2}

= π {(r + h)^{2} - r^{2 }}

= π (r + h - r) (r + h + r)

= π (2r + h) h

The area of a circle whose area and circumference are numerically equal, is

(a) 2π sq. units

(b) 4π sq. units

(c) 6π sq. units

(d) 8π sq. units

Correct option: (b)

area = circumference

πr^{2} = 2πr

r = 2 units

area = πr^{2}

= 4π sq. units

If diameter of a circle is increased by 40%, then its area increases by

(a) 96%

(b) 40%

(c) 80%

(d) 48%

In figure, the shaded area is

(a) 50 (π - 2) cm^{2}

(b) 25 (π - 2) cm^{2}

(c) 25 (π + 2) cm^{2}

(d) 5 (π - 2) cm^{2}

** img pending

## Chapter 13 - Areas Related to Circles Exercise 13.72

If the area of a sector of a circle bounded by an arc of length 5π cm is equal to 20π cm^{2}, then the radius of the circle is

(a) 12 cm

(b) 16 cm

(c) 8 cm

(d) 10 cm

In Figure, the ratio of the areas of two sectors S_{1 }and S_{2} is

(a) 5 : 2

(b) 3 : 5

(c) 5 : 3

(d) 4 : 5

## Chapter 13 - Areas Related to Circles Exercise 13.73

In figure, the area of the shaded region is

(a) 3π cm^{2}

(b) 6π cm^{2}

(c) 9π cm^{2}

(d) 7π cm^{2}

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

(a) 13 : 22

(b) 14 : 11

(c) 22 : 13

(d) 11 : 14

If a chord of a circle of radius 28 cm makes an angle of 90° at the centre, then the area of the major segment is

(a) 392 cm^{2}

(b) 1456 cm^{2}

(c) 1848 cm^{2}

(d) 2240 cm^{2}

If area of a circle inscribed in an equilateral triangle is 48π square units, then perimeter of the triangle is

## Chapter 13 - Areas Related to Circles Exercise 13.74

The hour hand of a clock is 6 cm long. The area swept by it between 11.20 am and 11.55 am is

(a) 2.75 cm^{2}

(b) 5.5 cm^{2}

(c) 11 cm^{2}

(d) 10 cm^{2}

If the area of circle is equal to the sum of the areas of two circles of diameters 10 cm and 24 cm, then diameter of the larger circle (in cm) is

(a) 34

(b) 26

(c) 17

(d) 14

Correct option: (b)

radius of Circle = 5 cm

area = π (5)^{2}

= 25 π

rdius of circle 2 = 12 cm

area = π (12)^{2}

= 144 π

area of larger circle = 144 π + 25π

= 169 π

πr^{2} = 169 π

r^{2} = 169

r = 13

diameter = 2r

= 26

If Π is taken as 22/7, the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution, is

(a) 2.2

(b) 1.1

(c) 9.625

(d) 96.25

ABCD is a rectangle whose three vertices are B(4, 0), C(4, 3) and D(0, 3). The length of one of its diagonals is

(a) 5

(b) 4

(c) 3

(d) 25

Area of the largest triangle that can be inscribed in a semi-circle of a radius r units is

a. r^{2} sq. units

b.

c. 2r^{2} sq. units

d.

If the sum of the areas of two circles with radii r_{1} and r_{2} is equal to the area of a circle of radius r, then

a. r = r_{1} + r_{2}

b.

c. r_{1} + r_{2} < r

d.

If the sum of the circumference of two circles with radii r_{1} and r_{2} is equal to the circumference of a circle of radius r, then

a. r = r_{1} + r_{2}

b. r_{1} + r_{2} > r

c. r_{1} + r_{2} < 2

d. None of these

If the circumference of a circle and the perimeter of a square are equal, then

a. Area of the circle = Area of the square

b. Area of the circle < Area of the square

c. Area of the circle > Area of the square

d. Nothing definite can be said

If the perimeter of a circle is equal to that of a square, then the ratio of their areas is

a. 22 : 7

b. 14 : 11

c. 7 : 22

d. 11 : 14

## Chapter 15 - Areas Related to Circles Exercise 15.68

Correct Option (b)

## Chapter 15 - Areas Related to Circles Exercise 15.69

Correct option - (c)

### Other Chapters for CBSE Class 10 Mathematics

Chapter 1- Real Numbers Chapter 2- Polynomials Chapter 3- Pairs of Linear Equations in Two Variables Chapter 4- Quadratic Equations Chapter 5- Arithmetic Progressions Chapter 6- Co-ordinate Geometry Chapter 7- Triangles Chapter 8- Circles Chapter 9- Constructions Chapter 10- Trigonometric Ratios Chapter 11- Trigonometric Identities Chapter 12- Heights and Distances Chapter 14- Surface Areas and Volumes Chapter 15- Statistics Chapter 16- Probability### RD SHARMA Solutions for CBSE Class 10 Subjects

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