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61. The perimeter of a semi circle of 56 cm diameter will be:
  A.  151 cm
  B.  144 cm
  C.  121 cm
  D.  160 cm
     
   
View Answer

Shortcut:
To find the circumference of a circle when radius is given.
Circumference of a circle = 2πr or πd [where Diameter (d) = 2r]
Note: To find the radius of a circle when perimeter or circumference is given.
(i) Radius of a circle =
Perimeter or circumference / 2π

(ii) Diameter =
Perimeter / π

Here, d = 56
Using these values in the shortcut, we get:
Perimeter = (πr + 2r) =
22 / 7
x 28 + 56
= (4 x 22) + 56 = 88 + 56 = 144
Hence, the perimeter is 144 cm.

62. A circular wire of radius 42 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 6 : 5. The smaller side of the rectangle is:
  A.  60 cm
  B.  50 cm
  C.  52 cm
  D.  45 cm
     
   
View Answer

Shortcut:
To find the circumference of a circle when radius is given.
Circumference of a circle = 2πr or πd [where Diameter (d) = 2r]
Note: To find the radius of a circle when perimeter or circumference is given.
(i) Radius of a circle =
Perimeter or circumference / 2π

(ii) Diameter =
Perimeter / π

Here, r = 42
Perimeter of circular wire = Perimeter of rectangle
2πr= 2(l + b)
Let l = 6y & b = 5y
Then, 2 x
22 / 7
x 42 = 2(6y + 5y)
or, 22 x 6 = 11y
or, y = 2 x 6 = 12
∴ smaller side = 5y = 5 x 12 = 60
Hence, the smaller side is 60 cm.

63. The length of a rope by which a buffalo must be tethered in order that she may be able to graze an area of 9856 sq m, is:
  A.  42 m
  B.  52 m
  C.  45 m
  D.  56 m
     
   
View Answer

Shortcut:
To find the area of a circle when radius is given.
Area of a circle = πr2 = π x
(
Diameter / 2
)
2 [Diameter = 2 x radius]
Note: To find the radius of a circle when its area is given.
(i) Radius of a circle =
√(
Area / π
)

(ii) Diameter of the circle = 2 x
√(
Area / π
)

Here, Area = 9856, r = ?
Using these values in the shortcut, we get:
Area = πr2
9856 =
22 / 7
x r2
⇒ r2 =
9856 x 7 / 22

r2 = 3136
r = 56
Hence, the length of a rope is 56 m.

64. The area of a triangular plate of which the base and the altitude are 33 cm and 14 cm respectively is to be reduced to one third by drilling a circular hole throught it. Calculate the diameter of the hole.
  A.  16 cm
  B.  11 cm
  C.  14 cm
  D.  10 cm
     
   
View Answer

Shortcut:
To find the area of a circle when radius is given.
Area of a circle = πr2 = π x
(
Diameter / 2
)
2 [Diameter = 2 x radius]
Note: To find the radius of a circle when its area is given.
(i) Radius of a circle =
√(
Area / π
)

(ii) Diameter of the circle = 2 x
√(
Area / π
)

Here, b = 33, h = 14
Area of triangular plate =
1 / 2
x b x h
=
1 / 2
x 33 x 14 = 231
Reduced area =
1 / 3
x 231 = 77
Area of hole = 231 − 77 = 154
πr2 = 154
or, r2 =
154 x 7 / 22
= 7 x 7
r = 7
D = 2r = 2 x 7 = 14
Hence, the diameter of hole 14 cm.

65. The circumference of a circle is 352 m. Its area is:
  A.  9860 sq m
  B.  9856 sq m
  C.  9836 sq m
  D.  9956 sq m
     
   
View Answer

Shortcut:
To find the area of a circle when perimeter or circumference is given.
Area of a circle = =
(circumference)2 / 4π

Here, circumference = 352
Area of a circle =
(352)2 / 4π

Area of a circle =
352 x 352 x 7 / 4 x 22
= 4 x 352 x 7 = 9856
Hence, the area of circle 9856 sq m.

66. In a circle of radius 28 cm, an arc subtends an angle of 72° at the centre. Find the length of the arc and the area of the sector so formed.
  A.  33.4 cm; 490.4 sq cm
  B.  36.2 cm; 488.2 sq cm
  C.  35.2 cm; 492.8 sq cm
  D.  32.8 cm; 498.2 sq cm
     
   
View Answer

Shortcut:
To find arc of a sector, when θ (angle subtended by the arc at the centre of a circle of which arc is a part) and circumference ( or perimeter) is given.
(i) Area of a sector = circumference x
θ / 360

(ii) Circumference =
360 x Arc of sector / θ

(iii) Area of a sector =
θ / 360
x πr2
[If only radius (r) of the circle is given, then]
Area of a sector =
1 / 2
x radius x length of arc
Here, r = 28, θ = 72
Lenght of arc =
2πr x θ / 360

= 2 x
22 / 7
x 28 x
72 / 360
= 35.2 cm
Area of the sector =
πr2 x θ / 360

=
22 / 7
x 28 x 28 x
72 / 360
= 492.8 sq cm

67. If a piece of wire 20 cm long is bent into an arc of a circle subtending an angle of 60° at the centre, then the radius of the circle is:
  A.  
75 / π
cm
  B.  
55 / π
cm
  C.  
45 / π
cm
  D.  
60 / π
cm
     
   
View Answer

Shortcut:
To find arc of a sector, when θ (angle subtended by the arc at the centre of a circle of which arc is a part) and circumference ( or perimeter) is given.
(i) Area of a sector = circumference x
θ / 360

(ii) Circumference =
360 x Arc of sector / θ

(iii) Area of a sector =
θ / 360
x πr2
[If only radius (r) of the circle is given, then]
Area of a sector =
1 / 2
x radius x length of arc
Here, circumference = 20, θ = 60, r = ?
Using these values in the shortcut, we get:
2 x π x r =
360 x 20 / 60

r =
6 x 20 / 2 x π
=
60 / π

Hence, the radius of the circle is
60 / π
cm.

68. A horse is placed inside a rectangular enclosure 40 m by 36 m and is tethered to one corner by a rope 14 m long.
(i) Over what area can it graze:
(ii) If the horse is outside the enclosure and is tethered to the corner by the same rope, over what area can it graze?
  A.  (i) 158 sq m; (ii) 465 sq m
  B.  (i) 154 sq m; (ii) 462 sq m
  C.  (i) 158 sq m; (ii) 450 sq m
  D.  (i) 144 sq m; (ii) 470 sq m
     
   
View Answer

Shortcut:
To find arc of a sector, when θ (angle subtended by the arc at the centre of a circle of which arc is a part) and circumference ( or perimeter) is given.

(i) Area of a sector = circumference x
θ / 360

(ii) Circumference =
360 x Arc of sector / θ

(iii) Area of a sector =
θ / 360
x πr2
[If only radius (r) of the circle is given, then]
Area of a sector =
1 / 2
x radius x length of arc
Here, OAB is a sector of a circle whose radius is 14 metres.
∴ Area of the sector OAB =
90 / 360
x
22 / 7
x 14 x 14
=
1 / 4
x 22 x 2 x 14 = 154 sq m
(ii) Required area =
22 / 7
x 14 x 14 − 154 = 616 − 154 = 462 sq m.

69. Find the length of the arcs cut off from a circle of radius 7 cm by a chord 7 cm long.
  A.  7
1 / 3
m; 36
2 / 3
m
  B.  5
3 / 7
m;35
3 / 7
m
  C.  3
1 / 7
m; 39
5 / 9
m
  D.  6
1 / 4
m; 35
7 / 3
m
     
   
View Answer

Shortcut:
To find arc of a sector, when θ (angle subtended by the arc at the centre of a circle of which arc is a part) and circumference ( or perimeter) is given.
(i) Area of a sector = circumference x
θ / 360

(ii) Circumference =
360 x Arc of sector / θ

(iii) Area of a sector =
θ / 360
x πr2
[If only radius (r) of the circle is given, then]
Area of a sector =
1 / 2
x radius x length of arc

Here, Two arcs will be cut off, one smaller and the other bigger.
Δ OAB is and equilateral triangle.
∴ ∠O = 60°
∴ Length of the arc ADB =
60 / 360
x 2 x
22 / 7
x 7
=
1 / 6
x 2 x 22 =
22 / 3
or 7
1 / 3
m
Length of the arc AEB = 2 x
22 / 7
x 7 −
22 / 7
=
110 / 3
= 36
2 / 3
m

70. The radius of a circle of centre O is 5 cm. Two radii OA and OB are drawn at right angles to each other. Find the areas of the two segments made by the chord BA.
  A.  6
3 / 2
sq m, 73
4 / 7
sq m
  B.  8
3 / 7
sq m,72
1 / 7
sq m
  C.  7
1 / 7
sq m, 71
3 / 7
sq m
  D.  5
3 / 7
sq m, 69
5 / 3
sq m
     
   
View Answer

Shortcut:
To find arc of a sector, when θ (angle subtended by the arc at the centre of a circle of which arc is a part) and circumference ( or perimeter) is given.
(i) Area of a sector = circumference x
θ / 360

(ii) Circumference =
360 x Arc of sector / θ

(iii) Area of a sector =
θ / 360
x πr2
[If only radius (r) of the circle is given, then]
Area of a sector =
1 / 2
x radius x length of arc

Here, Area of the sector OADB =
90 / 360
x
22 / 7
x 5 x 5 =
275 / 14
sq m
Area of the Δ OAB =
1 / 2
x 5 x5 =
25 / 2
sq m
∴ Area of the segment ADB =
275 / 14
−
25 / 2
=
50 / 7
= 7
1 / 7
sq m
Area of the segment AEB =
22 / 7
x 5 x 5 −
50 / 7
=
500 / 7
= 71
3 / 7
sq m.

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