# Solution of Chapter 15. Areas Related to Circles (RD Sharma - Mathematics Book)

## Exercise 15.1

1

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

2

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

3

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

4

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

5

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

6

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

7

A horse is placed for grazing inside a rectangular field 40m by 36m and is tethered to one corner by a rope 14m long. Over how much area can it graze?.

8

A sheet of paper is in the form of a rectangle ABCD in which AB=40cm and AD=28cm. A semi-circular portion with BC as diameter is cut off. Find the area of the remaining paper.

9

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

10

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

11

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

12

The area of a circle inscribed in an equilateral triangle is 154cm2. Find the perimeter of the triangle.

13

A field is in the form of a circle. A fence is to be erected around the field. The cost of fencing would be Rs.2640 at the rate of Rs.12 per metre. Then, the field is to be thoroughly ploughed at the cost of Re.0.50 per m2. What is the amount required to plough the field?

14

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

15

A park is in the form of a rectangle 120m×100m. At the centre of the park there is a circular lawn. The area of park excluding lawn is 8700m2. Find the radius of the circular lawn.

16

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

17

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

18

A car travels 1 kilo meter distance in which each wheel makes 450 complete revolutions. Find the radius of its wheels.

19

The area enclosed between the concentric circles is 770cm2. If the radius of the outer circle is 21cm, find the radius of the inner circle.

## Exercise 15.2

1

Find, in terms of, the length of the arc that subtends an angle of 30° at the centre of a circle of radius 4cm.

2

Find the angle subtended at the centre of a circle of radius 5cm by an arc of length cm.

3

An arc of length 20π cm subtends an angle of 144° at the centre of a circle. Find the radius of the circle.

4

An arc of length 15cm subtends an angle of 45° at the centre of a circle. Find in terms of, the radius of the circle.

5

Find the angle subtended at the centre of a circle of a circle of radius ‘a’ by an arc of lengthcm.

6

A sector of a circle of radius 4cm contains an angle of 30°. Find the area of the sector.

8

The area of a sector of a circle of radius 2 cm iscm2. Find the angle contained by the sector.

9

The area of a sector of a circle of radius 5cm is 5cm2. Find the angle contained by the sector.

10

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

11

In a circle of radius 35cm, an arc subtends an angle of 72° at the centre. Find the length of the arc and area of the sector.

12

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

13

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

14

A sector is cut-off from a circle of radius 21cm. The angle of the sector is 120°. Find the length of its arc and the area.

15

The minute hand of a clock is cm long. Find the area described by the minute hand on the face of the clock between 7.00AM and 7.05AM.

16

The minute hand of a clock is 10cm long. Find the area of the face of the clock described by the minute hand between 8AM and 8.25AM.

17

A sector of 56° cut out from a circle contains area 4.4cm2. Find the radius of the circle.

18

In a circle of radius 6cm, a chord of length 10cm makes an angle of 110° at the centre of the circle. Find:

(i)the circumference of the circle,

(ii)the area of the circle,

(iii)the length of the arc AB,

(iv)the area of the sector OAB.

19

Fig.15.17, shows a sector of a circle, centre O, containing an angle θ°. Prove that:

(i) Perimeter of the shaded region is

(ii) Area of the shaded region is

20

Figure 15.18 shows a sector of a circle of radius r cm containing an angle θ°. The area of the sector is A cm2 and perimeter of the sector is 50cm.

(i)(ii)

21

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

22

In a circle of radius 21cm, an arc subtends an angle of 60° at the centre. Find (i)the length of the arc (ii)area of the sector formed by the arc

## Exercise 15.3

1

AB is a chord of a circle with centre O and radius 4cm. AB is of length 4cm and divides the circle into two segments. Find the area of the minor segment.

1

AB is a chord of a circle with centre O and radius 4cm. AB is of length 4cm and divides the circle into two segments. Find the area of the minor segment.

2

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

3

A chord of a circle of radius 14cm makes a right angle at the centre. Find the areas of the minor and major segments of the circle.

4

A chord 10cm long is drawn in a circle whose radius is cm. Find area of both the segments.

5

A chord AB of a circle, of radius 14 cm makes an angle of 60° at the centre of the circle. Find the area of the minor segment of the circle.

## Exercise 15.4

1

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

2

A play ground 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 play ground..

3

The outer circumference of a circular race-track is 525m. The track is everywhere 14m wide. Calculate the cost of leveling the track at the rate of 50paise per square meter

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.

5

Four equal circles, each of radius 5cm, touch each other as showing fig.15.65. Find the area included between them.

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?

8

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

9

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

10

A rectangular park is 100m by 50m. It is surrounded by semi-circular flower bed sall round. Find the cost of leveling the semi-circular flower bed sall 60paise per square meter.

11

Prove that the area of a circular path of uniform width h surrounding a circular region of radius is.

12

The inside perimeter of a running track (showninFig.15.67) is 400m. The length of each of the straight portion is 90m and the ends are semi-circles. If the track is everywhere 14m wide, find the area of the track. Also find the length of the outer running track.

13

Find the area of Fig15.68, in square cm, correct to one place of decimal.

14

In Fig.15.69, AB and CD are two diameters of a circle perpendicular to each other and OD is the diameter of the smaller circle. If OA=7cm, find the area of the shaded region.

15

In Fig.15.70, OACB is a quadrant of a circle with centre O and radius 3.5cm. If OD=2cm, find the area of the (i) quadrant OACB (ii) shaded region.

16

From each of the two opposite corners of a square of side 8cm, a quadrant of a circle of radius 1.4cm is cut. Another circle of radius 4.2cm is also cut from the centre as shown in Fig.15.71. Find the area of the remaining (shaded) portion of the square..

17

Find the area of the shaded region in Fig.15.72, if AC=24cm, BC=10cm and O is the centre of the circle.

18

In Fig.15.72(a), OABC is a square of side 7cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.

19

A circular pond is of diameter 17.5m. It is surrounded by a 2m wide path. Find the cost of constructing the path at the rate of ` 25 per square meter

20

A regular hexagon is inscribed in a circle. If the area of hexagon is,find the area of the circle.

21

A path of width 3.5m runs around a semi-circular grassy plot whose perimeter is 72m. find the area of the path.