Areas related to circles - Area of sector and segment of a circle (Class 10 maths)

Areas Related To Circles

Problems based onArea of sector and segment of a circle

Chapter : Areas Related To Circles Website: www.letstute.com


Q) A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find:a) The area of that part of the field in which the horse can grazeb) The increase in the grazing area if the rope were 10 m long

instead of 5 mUse [ π = 3.14]

Given: Side of a square = 15m Length of the rope = 5 m

To Find: a) Area of that part of the field in which the horse can graze. b) Increase in the grazing area if 10 m long rope is used


C

15m

15m

5 m

5m

A B

D

Q 15m

15m

P

10 m Q



Solution: a) Let ABCD represent the square shaped grass field of side 15 m.

Let D be the corner to which the horse is tethered and let DP (= 5m) be the rope by which it is tied.

Area of the part of the field over which the horse can graze = Area of quadrant DPQ

Thus, radius r (= 5m) and sector angle θ (= 90⁰)

C

15m

15m

5 m

5m

A B

D

Q 15m

15m

P

10 m Q



= θ x πr2 360

=

= 19.625 m2

Hence, the area of the part of the field in which the horse can graze is 19.625 m2

5514.3360

90m2

5514.34

1 = m2

4

5.78 = m2

Area of quadrant DPQ

C

15m

15m

5 m

5m

A B

D

Q 15m

15m

P

10 m Q



b) If r = 10 m, then the grazing area

= m2

= 78.5 m2

= m2

101014.3360

90

= θ x πr2 360

101014.34

1

= m2

4

314

C

15m

15m

5 m

5m

A B

D

Q 15m

15m

P

10 m Q



Increase in grazing area = (78.5 - 19.625) m2

=

Hence, the increase in the grazing area is 58.875 m2

58.875m2

C

15m

15m

5 m

5m

A B

D

Q 15m

15m

P

10 m Q



Q) An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

Given: Number of ribs = 8 Radius = 45 cm

To find: Area between two consecutive ribs = ?



Solution: Angle made by the two consecutive ribs of the umbrella at the centre

= Angle of the full circle Number of ribs

= 360⁰ 8= 45⁰



Area between two consecutive ribs

= Area of a sector of the circle of radius r (=45cm) and sector angle θ (=450)

= θ x πr2 360 = cm2

= cm2

4545

7

22

360

45

= cm2

4545

7

11

4

1

28

22275



Hence, the area between two consecutive ribs of the umbrella is 795.53 cm2

= 795.53 cm2



Q) The diagram represents the area swept by the wiper of a car with the dimensions given in the figure, calculate the shaded area swept by the wiper.

O

300

7 cm

14 cm

B

D

C

A

Given: OD 7 cm DC 14 cm

∠𝐂𝐎𝐁=∠𝐃𝐎𝐀=𝛉=𝟑𝟎𝟎

To find: Area swept by the wiper??



Solution: Let the radii of the sectors COB and DOA be ‘R’ and ‘r’ respectively

Then, R = (7+14) cm = 21 cm and r = 7 cm -------- (1)

300

7 cm

14 cm

O

B

D

C

A



300

O

B

D

C

A

Area of the shaded = [Area of sector COB – Area of sector DOA]region

= θ x πR2 - θ x πr2 360 360 = θ x π (R2 - r2) 360

= 30 x 22 (212 – 72) cm2 360 7

Using (1) and (2), we get,

= 1 x 22 (441 - 49) cm2

12 7

= 22 x 392 cm2

84

7 cm

14 cm



300 7 cm

14 cm

O

B

D

C

A

= 11 x 392 cm2

42

Hence, the area swept by the wiper is 102.67 cm2

102.67cm2

= 11 x 56 cm2

6

= 11 x 28 cm2

3

= 308 cm2

3

=



Q) In the given figure, O is the center of the concentric circles. Radius of the inner circle is half the radius of the outer circle. GivenAOC = 1350 and OA = 14 cm, Calculate the area of the shaded region. (Leave your answer in terms of π)

Given:AOC = 1350

OA = 14 cm

To find: Area of the shaded region = ?

S R

O1350

A D

C B

P Q

14 cm



Solution: Let the radii of the outer circle and inner concentric circles be ‘R’ and ‘r’ respectively.

Then, R = 14 cm and r = 7 cm -------- (1)

∠𝐀𝐎𝐃=𝟏𝟖𝟎𝟎−𝟏𝟑𝟓𝟎

= 450 ------- (Linear pair)

BOC = AOD = 450 (=θ) ----- V. opp.’s ---------- (2) S R

O1350

A D

C B

P Q

14 cm



Area of the shaded region 2[Area of the sector AOD – Area of sector POQ] 2

2

By using (1) and (2), we get,

¿𝟐×𝟓𝟒𝟎

𝛑 (𝟏𝟗𝟔−𝟒𝟗 )𝐜𝐦𝟐

cm2

¿𝟐×𝛉

𝟑𝟔𝟎×𝛑(𝐑𝟐−𝐫𝟐)

S R

O1350

A D

C B

P Q

14 cm



Result: Area =

Hence, the area of the shaded region is cm2

cm2

=

= cm2

S R

O1350

A D

C B

P Q

14 cm

19

Now we know…


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Areas related to circles - Area of sector and segment of a circle (Class 10 maths)