# Solution of Chapter 18. Area of Circle, Sector and Segment (RS Aggarwal - Mathematics Book)

## Exercise 18A

1

The difference between the circumference and radius of a circle is 37 cm. Using π = 22/7, find the circumference of the circle.

2

The circumference of a circle is 22 cm. Find the area of its quadrant.

3

What is the diameter of a circle whose area is equal to the sum of the areas of two circles of diameter 10 cm and 24 cm?

4

If the area of a circle is numerically equal to twice its circumference, then what is the diameter of the circle?

5

What is the perimeter of a square which circumscribes a circle of radius a cm?

6

Find the length of the arc of a circle of diameter 42 cm which subtends an angle of 60° at the centre.

7

Find the diameter of the circle whose area is equal to the sum of the areas of two circles having radii 4 cm and 3 cm.

8

Find the area of a circle whose circumference is 8π.

9

Find the perimeter of a semicircular protractor whose diameter is 14 cm.

10

Find the radius of a circle whose perimeter and area are numerically equal.

11

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

12

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

13

Find the area of the sector of a circle having radius 6 cm and of angle 30°.

14

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

15

The circumferences of two circles are in the ratio 2:3. What is the ratio between their areas?

16

The areas of two circles are in the ratio 4:9. What is the ratio between their circumferences?

17

A square is inscribed in a circle. Find the ratio of the areas of the circle and the square.

18

The circumference of a circle is 8 cm. Find the area of the sector whose central angle is 72°.

19

A pendulum swings through an angle of 30° and describes an arc 8.8 cm in length. Find the length of the pendulum.

20

The minute hand of a clock is 15 cm long. Calculate the area swept by it in 20 minutes.

21

A sector of 56°, cut out from a circle, contains 17.6 cm2. Find the radius of the circle.

22

The area of the sector of a circle of radius 10.5 cm is 69.3 cm2. Find the central angle of the sector.

23

The perimeter of a certain sector of a circle of radius 6.5 cm is 31 cm. Find the area of sector.

24

The radius of a circle is 17.5 cm. Find the area of the sector enclosed by two radii and an arc 44 cm in length.

25

Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular cardboard of dimensions 14 cm × 7 cm. Find the area of the remaining cardboard.

26

In the given figure, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is also drawn. Find the area of the shaded region. 27

From a rectangular sheet of paper ABCD wit AB = cm and AD = 28 cm, a semicircular portion wit BC as diameter is cut off. Find the area of the remaining paper.

28

In the given figure, OABC is a square of side 7 cm. If COPB is a quadrant of a circle wit centre C find the area of the shaded region. 29

In the given figure, three sectors of a circle of radius 7 cm, making angles of 60°, 80° and 40° at the centre are shaded. Find the area of the shaded region. 30

In the given figure, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm with centre O. If POQ = 30°, find the area of the shaded region. 31

In the given figure, find the area of the shaded region, if ABCD is a square of side 14 cm and APD and BPC are semicircle. 32

In the given figure, the shape of the top of a table is that of a sector of circle with centre O and AOB = 90°. If AO = OB = 42 cm, then find the perimeter of the top of the table. 33

In the given figure, ABCD is a square of side 7 cm, DPBA and DQBC are quadrants of circles each of the radius 7 cm. Find the area of shaded region. 34

In the given figure, OABC is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region. 35

Find the perimeter of shaded region in the figure, if ABCD is a square of side 14 cm and APB and CPD are semicircles. 36

In a circle of radius 7 cm, a square ABCD is inscribed. Find the area of the circle which is outside the square.

37

In the given figure, APB and CQD are semicircle of diameter 7 cm each, while ARC and BSD are semicircles of diameter 14 cm each. Find the (i) perimeter, (ii) area of the shaded region. 38

In the given figure, PSR, RTQ and PAQ are three semicircles of diameter 10 cm, 3 cm and 7 cm respectively. Find the perimeter of shaded region. 39

In the given figure, a square OABC is inscribed in a quadrant OPBQ of a circle. IF OA = 20 cm, find the area of the shaded region. [Use π = 3.14] 40

In the given figure, APB and AQO are semicircles and AO = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region. 41

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

42

In the given figure, find the area of the shaded region, where ABCD is a square of side 14 cm and all circles are of the same diameter. 43

Find the area of the shaded region in the given figure, if ABCD is a rectangle wit sides 8 cm and 6 cm ad O is the centre of the circle. 44

A wire is bent to form a square enclosing an area of 484 m2. Using the same wire, a circle is formed. Find the area of the circle.

45

A square ABCD is inscribed in a circle of radius ‘r’. Find the area of the square.

46

The cost of fencing a circular field at the rate of Rs. 25 per meter is Rs. 5500. The field is to be ploughed at the rate of 50 paise per m2. Find the cost of ploughing the field. [Take π = 22/7]

47

A park is in the form of a rectangle 120 m by 90 m. At the centre of the park, there is a circular lawn as shown in the figure. The area of the park excluding the lawn is 2950 m2. Find the radius of the circular lawn. [Given, π = 3.14] 48

In the given figure PQSR represents a flower be. If OP = 21 m and OR = 14 m, find the area of the flower bed. 49

In the given figure, O is the centre of the bigger circle, and AC is its diameter. Another circle with AB as diameter is drawn. If AC = 54 cm and BC = 10 cm, find the area of the shaded region. 50

From a thin metallic piece in the shape of a trapezium ABCD in which AB CD and BCD = 90°, a quarter circle BFEC is removed. Given, AB = BC = 3.5 cm and DE = 2 cm, calculate the area of remaining (shaded) part of metal sheet. 51

Find the area of the major segment APB of a circle of radius 35 cm and AOB = 90°, as shown in the given figure. ## Exercise 18B

1

The circumference of a circle is 39.6 cm. Find its area.

2

The area of a circle is 98.56 cm2. Find its circumference.

3

The circumference of a circle exceeds its diameter by 45 cm. Find the circumference of the circle.

4

A copper wire when bent in the form of a square encloses an area of 484 cm2. The same wire is now bent in the form of a circle. Find the area enclosed by the circle.

5

A wire when bent in the form of an equilateral triangle endoses an area of 121√3 cm2. The same wire is bent to form a circle. Find the area enclosed by the circle.

6

The length of a chain used as the boundary of a semicircular park is 108 m. Find the area of the park.

7

The sum of the radii of two circles is 7 cm, and the difference of their circumferences is 8 cm. Find the circumferences of the circles.

8

Find the area of a ring whose outer and inner radii are respectively 23 cm and 12 cm.

9

A path of 8 m width runs around the outside of a circular park whose radius is 17 m. Find the area of the path.

10

A racetrack is in the form of a ring whose inner circumference is 352 m and outer circumference is 396 m. Find the width and the area of the track.

11

A sector is cut from a circle of radius 21 cm. The angle of the sector is 150°. Find the length of the arc and the area of the sector.

12

The area of the sector of a circle of radius 10.5 cm is 69.3 cm2. Find the central angle of the sector.

13

The length of an arc of a circle, subtending an angle of 54° at the centre is 16.5 cm. Calculate the radius, circumference and area of the circle.

14

The radius of a circle with centre O is 7 cm. Two radii OA and OB are drawn at right angles to each other. Find the areas of minor and major segments.

15

Find the lengths of the arcs cut off from a circle of radius 12 cm by a chord 12 cm long. Also, find the area of the minor segment. [Take π = 3.14 and √3 = 1.73.]

16

A chord 10 cm long is drawn in a circle whose radius is 5√2 cm. Find the areas of both the segments. [Take π = 3.14.]

17

Find the area of both the segments of a circle of radius 42 cm with central angle 120°. [Given, sin 120° = √3/2 and √3 = 1.73.]

18

A chord of a circle of radius 30 cm makes an angle of 60° at the centre of the circle. Find the areas of the minor and major segments. [Take π = 3.14 and √3 = 1.732.]

19

In a circle of radius 10.5 cm, the minor arc is one-fifth of the major arc. Find the area of the sector corresponding to the major arc.

20

The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 2 days. [Take π = 3.14.]

21

Find the area of a quadrant of a circle whose circumference is 88 cm.

22

A rope by which a cow is tethered is increased from 16 m to 23 m. How much additional ground does it have now to graze?

23

A horse is placed for grazing inside a rectangular field 70 m by 52 m. It is tethered to one corner by a rope 21 m long. On how much area can it graze? How much area is left ungrazed?

24

A horse is tethered to one corner of a field which is in the shape of an equilateral triangle of side 12 m. If the length of the rope is 7 m, find the area of the field which the horse cannot graze. Take √3 = 1.732. Write the answer correct to 2 places of decimal.

25

Four cows are tethered at the four corners of a square field of side 50 m such that each can graze the maximum unshared area. What area will be left ungrazed? [Take π = 3.14.]

26

In the given figure, OPQR is a rhombus, three of whose vertices lie on a circle with centre O. If the area of the rhombus is 32√3 cm2, find the radius of the circle. 27

The side of a square is 10 cm. Find

(i) The area of the inscribed circle, and

(ii) The area of the circumscribed circle. [Take π = 3.14.]

28

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

29

The area of a circle inscribed in an equilateral triangle is 154 cm2. Find the perimeter of the triangle. [Take √3 = 1.73.]

30

The radius of the wheel of a vehicle is 42 cm. How many revolutions will it complete in a 19.8-km-long journey?

31

The wheels of the locomotive of a train are 2.1 m in radius. They make 75 revolutions in one minute. Find the speed of the train in km per hour.

32

The wheels of a car make 2500 revolutions in covering a distance of 4.95 km. Find the diameter of a wheel.

33

A boy is cycling in such a way that the wheels of his bicycle are making 140 revolutions per minute. If the diameter of a wheel is 60 cm, calculate the speed (in km/h) at which the boy is cycling.

34

The diameter of the wheels of a bus is 140 cm. How many revolutions per minute do the wheels make when the bus is moving at a speed of 72.6 km per hour?

35

The diameters of the front and rear wheels of a tractor are 80 cm and 2 m respectively. Find the number of revolutions that a rear wheel makes to cover the distance which the front wheel covers in 800 revolutions.

36

Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm. 37

Four equal circles, each of radius 5 cm, touch each other, as shown in the figure. Find the area included between them. [Take π = 3.14] 38

Four equal circles, each of radius a units, touch each other. Show that the area between them is sq units.

39

Three equal circles, each of radius 6 cm, touch one another as shown in the figure. Find the area enclosed between them. [Take π = 3.14 and √3 = 1.732.]

40

If three circles of radius a each, are drawn such that each touches the 4 other two, prove that the area included between them is equal to [Take √3 = 1.73 and π = 3.14.]

41

In the given figure, ABCD is a trapezium of area 24.5 cm2. If AD || BC, DAB = 90°, AD = 10 cm, BC = 4 cm and ABE is quadrant of a circle then find the area of the shaded region. 42

ABCD is a field in the shape of a trapezium, AD || BC, ABC = 90° and ADC = 60°. Four sectors are formed with centres A, B, C and D, as shown in the figure. The radius of each sector is 14 m. Find the following:

(i) total area of the four sectors,

(ii) area of the remaining portion, given that AD = 55 m, BC = 45 m and AB = 30 m.

43

Find the area of the shaded region in the given figure, where a circular arc of radius 6 cm has been drawn with vertex of an equilateral triangle of side 12 cm as centre and a sector of circle of radius 6 cm with centre B is made. [Use √3 = 1.73 and π = 3.14.] 44

In the given figure, ABCD is a rectangle with AB = 80 cm and BC = 70 cm, AED = 90° and DE = 42 cm. A semicircle is drawn, taking BC as diameter. Find the area of the shaded region. 45

In the given figure, from a rectangular region ABCD with AB = 20 cm, a right triangle 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. [Use π = 3.14.] 46

In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and BOD = 90°. Find the area of shaded region. [Use π = 3.14.] 47

In the given figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region. [Use √3 = 1.73 and π = 3.14.] 48

On a circular table cover of radius 42 cm, a design is formed by a girl leaving an equilateral triangle ABC in the middle, as shown in the figure. Find the covered area of the design. [Use √3 = 1.73] 49

The perimeter of the quadrant of a circle is 25 cm. Find its area.

50

A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. [Use π = 3.14.]

51

The radius of a circular garden is 100 m. There is a road 10 m wide, running all around it. Find the area of the road and the cost of levelling it at Rs. 20 per m2. [Use π = 3.14.]

52

The area of an equilateral triangle is 49√3 cm2. Taking each angular point as centre, circles are drawn with radius equal to half the length of the side of the triangle. Find the area of the triangle not included in the circles. [Take √3 = 1.73.]

53

A child draws the figure of an aeroplane as shown. Here, the wings ABCD and FGHI are parallelograms, the tail DEF is an isosceles triangle, the cockpit CKI is a semicircle and CDFI is a square. In the given figure, BP CD, HQ FI and EL DF. If CD = 8 cm, BP = HQ = 4 cm and DE = EF = 5 cm, find the area of the whole figure. [Take π = 3.14.] 54

A circular disc of radius 6 cm is divided into three sectors with central angles 90°, 120° and 150°. What part of the whole circle is the sector with central angle150°? Also, calculate the ratio of the areas of the three sectors.

55

A round table cover has six equal designs as shown in the given figure. If the radius of the cover is 35 cm then find the total area of the design. [Use √3 = 1.732 and π = 3.14.] 56

In the given figure, PQ = 24 cm, PR = 7 cm and 0 is the centre of the circle. Find the area of the shaded region. [Take π = 3.14.] 57

In the given figure, ΔABC is right-angled at A. Find the area of the shaded region if AB = 6 cm, BC = 10 cm and 0 is the centre of the incircle of ΔABC. [Take π = 3.14.] 58

In the given figure, ΔABC is right-angled at A. Semicircles are drawn on AB, AC and BC as diameters. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region. 59

PQRS is a diameter of a circle of radius 6 cm. The lengths PQ, QR and RS are equal. Semicircles are drawn with PQ and QS as diameters, as shown in the given figure. If PS = 12 cm, find the perimeter and area of the shaded region. [Take π = 3.14.] 60

The inside perimeter of a running track shown in the figure is 400 m. The length of each of the straight portions is 90 m, and the ends are semicircles. If the track is 14 m wide everywhere, find the area of the track. Also, find the length of the outer boundary of the track. ## Multiple Choice Questions (MCQ)

1

The area of a circle is 38.5 cm2. The circumference of the circle is

2

The area of a circle is 49π cm2. Its circumference is

3

The difference between the circumference and radius of a circle is 37 cm. The area of the circle is

4

The perimeter of a circular field is 242 m. The area of the field is

5

On increasing the diameter of a circle by 40%, its area will be increased by

6

On decreasing the radius of a circle by 30%, its area is decreased by

7

The area of a square is the same as the area of a circle. Their perimeters are in the ratio

8

The circumference of a circle is equal to the sum of the circumferences of two circles having diameters 36 cm and 20 cm. The radius of the new circle is

9

The area of a circle is equal to the sum of the areas of two circles of radii 24 cm and 7 cm. The diameter of the new circle is

10

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

11

If the sum of the areas of two circles with radii R1 and R2 is equal to the area of a circle of radius R then

12

If the sum of the circumferences of two circles with radii R1 and R2 is equal to the circumference of a circle of radius R then

13

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

14

The radii of two concentric circles are 19 cm and 16 cm respectively. The area of the ring enclosed by these circles is

15

The areas of two concentric circles are 1386 cm2 and 962.5 cm2. The width of the ring is

16

The circumferences of two circles are in the ratio 3 : 4. The ratio of their areas is

17

The areas of two circles are in the ratio 9: 4. The ratio of their circumferences is

18

The radius of a wheel is 0.25 m. How many revolutions will it make in covering 11 km?

19

The diameter of a wheel is 40 cm. How many revolutions will it make in covering 176 m?

20

In making 1000 revolutions, a wheel covers 88 km. The diameter of the wheel is

21

The area of a sector of angle θ° of a circle with radius R is

22

The length of an arc of a sector of angle θ° of a circle with radius R is

23

The length of the minute hand of a clock is 21 cm. The area swept by the minute hand in 10 minutes is

24

A chord of a circle of radius 10 cm subtends a right angle at the centre. The area of the minor segments (given, π = 3.14) is

25

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

26

In a circle of radius 14 cm, an arc subtends an angle of 120° at the centre. If √3 = 1.73 then the area of the segment of the circle is

## Formative Assessment (Unit Test)

1

In the given figure, a square OABC has been inscribed in the quadrant OPBQ. If OA = 20 cm then the area of the shaded region is [take π = 3.14] 2

The diameter of a wheel is 84 cm. How many revolutions will it make to cover 792 m?

3

The area of a sector of a circle with radius r, making an angle of x° at the centre is x

4

In the given figure, ABCD is a rectangle inscribed in a circle having length 8 cm and breadth 6 cm. If π = 3.14 then the area of the shaded region is 5

The circumference of a circle is 22 cm. Find its area. [Take π = 22/7]

6

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

7

The minute hand of a clock is 12 cm long. Find the area swept by it in 35 minutes.

8

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

9

A chord of a circle of radius 14 cm makes a right angle at the centre. Find the area of the sector.

10

In the give figure, the sectors of two concentric circles of radii 7 cm and 3.5 cm are shown. Find the area of the shaded region. 11

A wire when bent in the form of an equilateral triangle encloses an area of 121√3 cm2. If the same wire is bent into the form of a circle, what will be the area of the circle? [Take π = 22/7]

12

The wheel of a cart is making 5 revolutions per second. If the diameter of the wheel is 84 cm, find its speed in km per hour. [Take π = 22/7]

13

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

(ii) the shaded region. [Take π = 22/7] 14

In the given figure, ABCD is a square each of whose sides measures 28 cm. Find the area of the shaded region. [Take π = 22/7] 15

In the given figure, an equilateral triangle has been inscribed in a circle of radius 4 cm. Find the area of the shaded region. [Take π = 3.14 and √3 = 1.73] 16

The minute hand of a clock is 7.5 cm long. Find the area of the face of the clock described by the minute hand in 56 minutes.

17

A racetrack is in the form of a ring whose inner circumference is 352 m and outer circumference is 396 m. Find the width and the area of the track.

18

A chord of a circle of radius 30 cm makes an angle of 60° at the centre of the circle. Find the areas of the minor and major segments. [Take π = 22/7 and √3 = 1.732.]