CHAPTER

11 Areas Related to Circles

TOPICS
Perimeter and Area of a circle.
Area of sector and segment of a circle.

Mathematics-X 151
KEY POINTS
Circle: A circle is the locus of a point which moves in a plane in such a way that
its distance from a fixed point always remains the same. The fixed point is
called the centre and the constant distance is known as the radius of the circle.
If r is radius of a circle, then
r2
(i) Area of semi circle =
2
r2
(ii) Area of quadrant of a circle =
4
(iii) If two circles touch internally, then the distance between their centres is
equal to the difference of their radii.
(iv) If two circles touch externally, then distance between their centres is equal
to the sum of their radii.
(v) Distance covered by rotating wheel in one revolution is equal to the
circumference of the wheel.
(vi) The number of revolutions completed by a rotating wheel in

Distance moved in one minute

one minute =
Circumference of the wheel

(vii) The sum of the arcs of major and minor sectors of a circle is equal to the
circumference of the circle.
(viii) The sum of the areas of major and minor sectors of a circle is equal to the
area of the circle.

VERY SHORT ANSWER QUESTIONS

1. If the diameter of a semi circular protactor is 14 cm, then find its perimeter.
2. If circumference and the area of a circle are numerically equal, find the diameter
of the circle.
3. Find the area of the circle ‘inscribed’ in a square of side a cm.
4. Find the area of a sector of a circle whose radius is r and length of the arc is l.
5. The radius of a wheel is 0.25 m. Find the number of revolutions it will make to
travel a distance of 11 kms.

152 Mathematics-X
6. If the area of a circle is 616 cm², then what is its circumference?
7. What is the area of the circle that can be inscribed in a square of side 6 cm?
8. What is the diameter of a circle whose area is equal to the sum of the areas of
two circles of radii 24 cm and 7 cm?
9. A wire can be bent in the form of a circle of radius 35 cm. If it is bent in the form
of a square, then what will be its area?
10. What is the angle subtended at the centre of a circle of radius 6 cm by an arc of
length 3 cm?
11. If the circumference of two circles are in the ratio 2:3, what is the ratio of their
areas?
12. If the difference between the circumference and radius of a circle is 37 cm, then
22
find the circumference of the circle. ( Use = )
7

13. If diameter of a circle is increased by 40%, find by how much percentage its
area increases?
14. The minute hand of a clock is 6 cm long. Find the area swept by it between
11:20 am and 11:55 am.
15. The perimeter of a sector of a circle of radius 14 cm is 68 cm. Find the area of
the sector. (CBSE 2020)
16. The circumference of a circle is 39.6 cm. Find its area.

22
(Use = ) (CBSE 2020)
7

17. The length of the minute hand of a clock is 14 cm. Find the area swept by the
22
minute hand in one minute. (Use = )
7

Mathematics-X 153
 MULTIPLE CHOICE QUESTIONS
18. 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
19. The Area of circle that can be inscribed in a square of side 6 cm is:
(a) 36 π cm² (b) 18 π cm²
(c) 12 π cm² (d) 9 π cm²
20. If the circumference of a circle increases from 4π to 8π, then Area is:
(a) Halved (b) Doubled
(c) Tripled (d) Quadrupled
21. If the perimeter of a semi- circular protractor is 36 cm , then its diameter is:
(a) 10 cm (b) 14 cm
(c) 12 cm (d) 16 cm
22. The length of a minute hand of clock is 14 cm. What is the area swept by the
mimute hand in 15 minutes?
(a) 154 cm² (b) 87 cm²
(c) 154 π cm² (d) 87 π cm²
23. The wheel of a cycle is of radius 35 cm. How many revolutions are required to
travels a distance of 11 m ?
(a) 2 (b) 5
(c) 10 (d) 15
24. Four horses are tied each with 7 m long rope at four corner of a square field of
sides 20 m. What is the area of field which can be grazed by the horses?
(a) 49 π m² (b) 98 π m²
(c) 74 π m² (d) 154 π m²

154 Mathematics-X
25. Find the area of a quadrant of a circle whose circumference is 22 cm.
22
(Use = )
7
26. What is the angle subtended at the centre of a circle of radius 10 cm by an arc of
length 5 cm?
27. If a square is inscribed in a circle, what is the ratio of the area of the circle and
the square?
28. Find the area of a circle whose circumference is 44 cm. (CBSE 2020)
29. If the perimeter of a circle is equal to that of square, then find the ratio of their
areas.
30. What is the ratio of the areas of a circle and an equilateral triangle whose
diameter and a side are respectively equal?
5
31. In figure, O is the centre of a circle. The area of sector OAPB is of the area
18
of the circle. Find x.

O
x
A B

P
32. Find the perimeter of the given figure, where AED is a semicircle and ABCD is a
rectangle. (CBSE 2015)
20 cm
B A
14 cm

C D
20 cm
33. In figure, OAPBO is a sector of a circle of radius 10.5 cm. Find the perimeter of
the sector.

Mathematics-X 155
 P

A B

60°

O
34. A Japenese fan can be made by sliding open its 7 small
sections, each of which is in the form of sector of a
circle having central angle of 15°. If the radius of this
fan is 24 cm, find the length of the lace that is required
to cover its entire boundary. (Use π = 22/7)
(CBSE 2014)
35. The perimeter of a sector of circle of radius 6.3 cm is 25.8 cm. Find the area of
the sector.
36. Find the area of a circle in which a square of area 64 cm2 is inscribed.
37. Find the area of a circle which is inscribed in a square of area 64 cm2.

SHORT ANSWER TYPE II QUESTIONS

38. Area of a sector of a circle of radius 36 cm is 54π cm2 . Find the length of the
corresponding arc of the sector.
39. The length of the minute hand of a clock is 5 cm. Find the area swept by the
minute hand during the time period 6:05 am to 6:40 am.
40. Find the area of the segment bounded by a chord AB and the arc ACB of the
circle with centre O having radius 7 cm and sector angle equal to 90°, as shown
in the figure.

156 Mathematics-X
41. In fig, OAPB is a sector of a circle of radius 3.5 cm with the centre at O and
∠AOB = 120°. Find the length of OAPBO.
P

120°
A B

42. Circular footpath of width 2 m is constructed at the rate of ` 20 per square

meter, around a circular park of radius 1500 m. Find the total cost of construction
of the foot path. (Take π = 3.14 )
43. A boy is cycling such that the wheels of the cycle are making 140 revolutions
per minute. If the diameter of the wheel is 60 cm. Calculate the speed of cycle.
44. In a circle with centre O and radius 4 cm, and of angle 30°. Find the area of
minor sector and major sector AOB. (Use π = 3.14)
45. Find the area of the largest triangle that can be inscribed in a semi circle of
radius r unit. (NCERT Exemplar)
46. In a square park of side 8 m two goats are tied at opposite vertices with a rope
of length 1.4 m and a cow is tied in the centre with a rope of length 2.1m.
Calculate the area of park which cannot be grazed by them.
47. A sector of 100° cut off from a circle contains area 70.65 cm². Find the radius of
the circle. (Use π = 3.14 )
48. The hour and minute hand of a 12 hour clock are 3.5 cm and 7 cm long
 22 
respectively. Find the sum of distance travelled by their tips in a day.  use π = 
 7 
49. A square water tank has its each 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 sq. m.
(Use π = 3.14 )
50. Length of a chord of a circle of a radius of 4 cm is 4 cm. Find the area of the
sector and segment formed by the chord.

Mathematics-X 157
51. Find the area of the minor segment of a circle of radius 21 cm, when the angle of
the corresponding sector is 120°.
52. A piece of wire 11 cm long is bent into the form of an arc of a circle subtending
an angle of 45° at its centre. Find the radius of the circle.
53. The circumference of a circle exceeds the diameter by 16.8 cm. Find the radius
of the circle.
54. A pendulum swings through an angle of 45° and describes an arc of 22 cm in
 22 
length. Find the length of the pendulum.  use π = 
 7 

LONG ANSWER TYPE QUESTIONS

55. Two circles touch externally. The sum of their areas is 130π sq. cm and the
distance between their centres is 14 cm. Find the radii of the circles.
56. Find the number of revolutions made by a circular wheel of area 6.16 m² in
rolling a distance of 572 m.
57. Three horses are tied at the vertices of a triangular park of sides 35 m, 84 m and
91 m with the help of a rope of length 14 m each. Calculate the ratio of the area
which can be grazed to the area which can’t be grazed.
58. Two circle touch each other internally. The sum of their area is 116π cm2 and
distance between their centres is 6 cm. Find the radii of the circles.
(CBSE = 2017)

ANSWERS AND HINTS

22
1. πr + d = × 7 + 14 = 36 cm
7
2. 2πr = πr2 ⇒ diameter = 4 units
3. Side of the square is equal to diameter of the circle,
a2 a
πr = π ×
2
(side = a, radius = )
4 2
θ θ l × πr 2 lr
4. l= × 2 π r , Area = × π r 2
= = sq. units
360° 360° 2 πr 2

158 Mathematics-X
 distance 11 × 1000 × 7 × 100
5. = = 7000
circumference 2 × 22 × 25
6. πr2 = 616 ⇒ r = 14 cm or 2πr = 88 cm
7. Side of the square is equal to the diameter of the circle
⇒ r = 3 cm or πr2 = π(3)2 = 9π cm2 .

8. πR 2 = πr12 + πr22 ⇒ R = 25 and diameter = 50 cm.

22 220
9. 2πr = 2 × × 35 = 220 cm , Side of square = 55 cm
7 4
Area of square = 55 × 55 = 3025 cm2

θ θ
10. l = × 2πr ⇒ 3π = × 2π× 6 ⇒ θ = 90°
360° 360°
2
2 
πr1  3 2 
r
2πr1 2 2 2
11. = ⇒ r1 = r2 or = = 4:9
2πr2 3 3 πr22 r22
22
12. (2πr – r) = 37 or r = 7, 2πr = 2 × × 7 = 44 cm
7
13. 96%
210°× 22 × 6 × 6
14. = 66 cm2 (θ = 210°) (11: 20 to 11: 55 = 35 minutes)
360°× 7
15. 280 cm2
16. 124.74 cm2
17. 10.27 cm2
18. (b) 14:11
19. (d) 9 π cm²
20. (d) Quadrupled
21. (d) 14 cm
22. (a) 154 cm²
23. (b) 5
24. (a) 49 π m²
Mathematics-X 159
 7
25. 2πr = 22, r =
2
πr 2 22 × 7 × 7
Area of quadrant = = = 9.625 cm2
4 7×4×2×2
θ θ
26. l = × 2πr ⇒ 5π = × 2π × 10 ⇒ θ = 90°
360° 360°

27.

If side of square is 1 unit, by Pythagoras Theorem

Diameter 2 unit.
Area of square = 1 × 1 = 1 sq units.
2 2 π 11
Area of Circle = πr 2 = π × × = =
2 2 2 7
Required ratio = 11 : 7
28. 154 cm2
2πr Perimeter of circle
29. 2πr = 4 unit or = (Let side of square = 1 unit)
4 unit Perimeter of square
7
r= unit
11
πr 2 22 7 7 14
= × × = or 14 : 11
1 7 11 11 11
3 2
30. Area of equilateral triangle = a
4
2
 a
Area of circle = π  
2
Required ratio = 3:π
θ 5
31. πr 2 = πr 2 ×
360° 18
θ = 100°
160 Mathematics-X
32. 20 cm + 14 cm + 20 cm + πr
22
20 cm + 14 cm + 20 cm + × 7 = 76 cm
7
θ 60 × 2 × 22 × 105
33. × 2πr = = 11 cm
360° 360°× 7 × 10
Perimeter = 10.5 + 10.5 + 11 cm = 32 cm
34. θ = 7 × 15° = 105°
θ
l= 2πr = 44 cm
360°
Length of lace = l + 2r
= 44 + 48 = 92 cm
35. Perimeter of sector = l + 2r
l = 25.8 – 12.6 = 13.2 cm
θ
× 2πr = l
360°
θ
Area of sector = πr 2
360°
Area of sector = 41.58 cm2
36. d = Diagonal of square
d = side 2 = 8 2 cm
O
r = 4 2 cm
Area = πR2 = 32π cm2

37. Diameter of circle = Side of square

∴ r = 4 cm
Area = 16π cm2
θ× π × 36 × 36
38. 54 π =
360°
θ = 15°
θ 15°× 2 × π× 36
l= × 2πr = = 3 π cm
360° 360°

Mathematics-X 161
 210 22 5 5 1650 5
39. Area = r2 45 cm2
360 360 7 36 6
( = 210° in 35 minutes)
40. Area of sector = area of sector – area of AOB
77 49
=
2 2
= 14 cm2
240 2 22 35
41. l=
360 7 10
= 14.67
Length of OAPBO = 14.6 + 3.5 + 3.5
= 21.67 cm

42. (r22 r12 ) = [(1502)2 – (1500)2 ] 20

= 3.14 [(1502)2 – (1500)2] × 20
= ` 377051.2
43. Circumference of cycle = 2 r
22
= 2 30 cm
7
= 188.57 cm
18857 140 60
Speed of cycle =
100 1000
= 15.84 km/h

44. Area of Minor sector = r2

360
30
= 3.14 4 4 cm 2
360
= 4.19 cm2 (approx.)

Area of major sector = r2

360

162 Mathematics-X
 330°
= × 3.14 × 4 × 4
360°
= 46.1 cm2 (approx)
1
45. Area of ∆ = base × height
2
1
= AB × OC
2
1
2r × r = r2 square unit
=
2
46. Grazing area of Goats = 2 × area of quadrants
22 1
= 2× × 1.4 × 1.4 × = 3.08m 2
7 4
Grazing area of cow = Ar. of circle
22
= × 2.1 × 2.1 = 13.86m 2
7
Area which can’t be grazed = Area of square – total grazing area
= 64 – 16.94 = 43.06 m2

7065 100°× 314 × r 2

47. =
100 360° × 100
7065 × 360
= r2
100 × 314
9= r
r = 9 cm.
48. Distance by minute hand in 1 day = 24 × 2πR
Distance by hour hand in 1 day = 2 × 2πr
Total distance travelled by tips of both hands = 24 × 2πR + 2 × 2πR
= 1056 + 44
= 1100 cm

49. Four semicircluar means 2 circles ,

Area of 2 circles = 2πr 2
= 2 × 3.14 × 20 × 20

Mathematics-X 163
 = 2512 sq.m
Total cost = 2512 × 1.25
= ` 3140
50. Length of chord = radius
∴ Angle of sector = 60°
θ
Area of sector = × πr 2
360°
8π 2
= cm
3
Area of segment = Area of sector - Area of triangle
8π 3 2
= – r
3 4

 8π 
=  − 4 3  cm
2

 3 

51. Area of the segment = Area of sector – Area of ∆

120° 22
Area of sector = × × 21× 21 = 462 cm2
360° 7
441
Area of ∆ = 3 cm 2
4
 441 
Area of segment =  462 − 3  cm2

4

=
21
4
(88 − 21 3 ) cm2
θ
52. l= × 2πr
360°
45° 2 × 22 × r
11 = ×
360° 7
14 = r
r = 14 cm

164 Mathematics-X
53. 2πr = 2r + 16.8

22 168  22  168
2× r − 2r = or 2r  − 1 =
7 10  7  10

 15  168 168 × 7 1176

or, 2r   = or r = = = 3.92 cm
 7 10 10 × 2 × 15 300

θ
54. l= × (2πr )
360°

45 22
22 = × 2× × r
360° 7
r = 28
⇒ Length of pendulum = 28 cm

55. πr12 + πr22 = 130 π ⇒ r12 + r22 = 130 ...(1)

⇒ r1 + r2 = 14 …(2)
Substitute the value of r1 from (2) in (1) and solve.
2r22 – 28 r2 + 66 = 0

r22 – 14r2 + 33 = 0 (Neglecting – ve)

r2 = 11 cm and r1 = 3 cm

616
56. πr2 = or r 2 = 1.96 or r = 1.4 m
100

22 14 616
2πr = 2 × × = = 8.8 m
7 10 100

572
Number of revolutions = = 65
8.8

180° 22
57. Grazing area of Horses = × × (14)2 = 308m 2
360° 7

Mathematics-X 165
 1
Area of triangular park = 35 84 1470m 2
2
Area which can’t be grazed = 1162m2
Grazing Area : Area can’t be grazed = 308 : 1162
= 22 : 83
58. R2 + r2 = 116 ...(1)
R–r=6 ...(2)
Squaring both sides and solving, we get
2Rr = 80 ...(3)
Addign and solving (1) and (3)
R + r = 14 ...(4)
Solving (2) and (4)
R = 10 cm, r = 4 cm

166 Mathematics-X
 PRACTICE-TEST
AREAS RELATED TO CIRCLES
Time : 45 Minutes M.M.: 20

SECTION-A
7
1. If the area of sector is of the area of the circle. Find the measure of central
18
angle of the sector. 1
2. The diameter of a circle whose area is equal to the sum of the areas of the two
circles of radii 24 cm and 7 cm is: 1
(a) 48 cm (b) 31 cm (c) 25 cm (d) 17 cm
3. The area of sector whose perimeter is four times its radius of measure r units
is_________. 1
4. If the area of a sector of a circle bounded by an arc of length 5 π cm is equal to
20 π cm2, then find the radius of the circle.

SECTION-B
5. The perimeter of a sector of circle of radius 5.7 cm is 27.2 cm. Find the area of
the sector. 2
6. The minute hand of a clock is 12 cm long. Find the area of the face of the clock
described by the minute hand between 6:10 pm and 6:45 pm. 2
7. Two circular pieces of equal radii and maximum area, touching each other are
cut out from a rectangular cardboard of dimensions 16 cm × 8 cm. Find the area
of the remaining cardboard. 2
SECTION-C
8. The length of a rope by which a cow is tied is increased from 12m to 19m. How
much more area can the cow graze now? (Use π = 22/7) 3
9. A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find
the area of the corresponding minor segment. (Use π = 22/7) 3

SECTION-D
10. Find the area of minor and major segments of a circle of radius 42 cm, if the
length of the arc is 88 cm. 4

Mathematics-X 167