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Templates covered in this chapter:
Template 1: Find the circumference of the circle when either radius or area of circle is given
...

Template 5: Find the area of minor and major sectors when angle of sector and radius are given
...

Steps required to follow:
Step 1: Find the radius of the circle from the given information:
If the radius of the circle is not given directly, then calculate the radius of the circle
Case 1: from diameter of the circle, if information of diameter is given
...

Step 2: Find the circumference of circle:
Substitute the value of radius in the formula of circumference of the circle and evaluate it
...
by multiplying the cost of fencing per unit length with circumference of circle
...


Question 1:
The area of a circle is 98
...
Find its circumference
...
56 𝑐𝑚2

… (1)

But
Area of circle = 𝜋𝑟 2
Therefore from eq (1) and (2)
𝜋𝑟 2 = 98
...
56
7

𝑟2 = (

7 × 98
...
92
22

)

)

𝑟 2 = 31
...
6 cm
Step 2: Find the circumference of circle:
We know that, Circumference of circle = 2 𝜋 𝑟
= 2 ( 3
...
6)

… (3)

[ Since 𝑟 = 5
...
168
Therefore, circumference of circle is 𝟑𝟓
...
Find the circumference of the circle
...
F
...

We know that,
Diameter of circle = 2𝑟
∴ Equation (1) becomes
C
...
C = 2𝑟 + 45
2𝑟 = 𝐶
...
𝐶−45

……………… (2)

2

Step 2: Find the circumference of circle:
Circumference of circle (C
...
C) = 2𝜋𝑟
Put
The value or radius r in equation (3)
C
...
C =

2𝜋(𝐶
...
𝐶 − 45 )

… (3)

……………………
...
𝐶 − 45)
7

7 (𝐶
...
𝐶) − 990
990 = ( 22 − 7 )( 𝐶
...
𝐶
therefore,
C
...
C =

990
15

C
...
C = 66 cm
Therefore, circumference of the circle is equal to 𝟔𝟔 cm
...

Steps required to follow:
Step 1: Find the radius of the circle from the given information:
If the radius of the circle is not given directly, then calculate the radius of the circle
Case 1: from diameter of the circle, if information of diameter is given
...

Step 2: Find the area of circle:
Substitute the value of radius in the formula of area of the circle and evaluate it
...
by multiplying the cost of painting per unit area with the area of the circle
...


Question 1:
The circumference of a circle is 39
...
Find its area
...
6 cm

……………
...
(2)

Now from Equation(1) and equation (2)

2𝜋𝑟 = 39
...
6
7

39
...
2
44

𝑟 = 12
...
6)2
7

= 498
...
96 𝑐𝑚2

Question 2:
Find the area of a circle whose circumference is 66 cm
...
(1)

= 2𝜋𝑟
Now from equation
...
(2)

Therefore the area of circle is 𝟏𝟑𝟖𝟔 𝒄𝒎𝟐
...

If radius of each circle is not given directly, find it when,
Case 1: Diameter of smaller circle or circumference of smaller circle and width of each circle is given
...

Case 2: Diameter of larger circle or circumference of larger circle and width of each circle is given
...

Step 2: Find the width of the circle:
If width of the circle is not given, but only circumference or radii of concentric circles are given
...
If width of ring/concentric circle is asked to find
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Question 1:
A road which is 7 m wide surrounds a circular park whose circumference is 352 m
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Solution:
Step 1: Identify the radius of the circle from the given information:
Let the inner radius of the park be 𝑟1 m and outer radius 𝑟2 m
...
(1)
but circumference of inner circle = 352 ………………(2)
Therefore, from equation (1) and equation (2)
2𝜋𝑟1 = 352
22

2 ( ) 𝑟1 = 352
7

𝑟1 = (
𝑟1 = (

7 × 352
44

2264
44

)

)

𝑟1 = 56 cm
But width of road = 7cm
Therefore, radius of outer circle is = 𝑟2 = 𝑟1 + 7
𝑟2 = 56 + 7
= 63 cm
Step 2: Find the area of each circle in the given concentric circle:
Required area of road = area of outer circle - area of inner circle
= 𝜋𝑟22 − 𝜋𝑟12
= 𝜋 (𝑟22 − 𝑟12 )
22

= ( ) (632 − 562 )
7

22

= ( ) (63 − 56 ) × (63 + 56)
7

22

= ( ) (7 × 119)
7

= (22) × (119)
Required area of road = 2618 𝑚2
Therefore, required area of road is 𝟐𝟔𝟏𝟖 𝒎𝟐

Question 2:
A race track is in the form of a ring whose inner and outer circumferences are 437 m and 503 m respectively
...


Solution:
Step 1: Identify the radius of the circle from the given information:
Let the inner radius of the park be 𝑟 m and outer radius 𝑅 m
...
(1)
but circumference = 437 ………………(2)
Therefore, from equation (1) and equation (2)
We get, 2𝜋𝑟1 = 437

2(3
...
28 )
𝑟1 = 69
...
5 m
Therefore, radius of inner circle is 𝟔𝟗
...

Similarly, we get circumference = 2𝜋𝑅 ………………
...

Step 2: Find the width of the concentric circle:
Now, Width of the track = 𝑅 − 𝑟
Width of the track = 80 − 69
...
5
Width of the track = 10
...
52 )
7

22

= ( ) × ( 80 − 69
...
5 )
7

22

= ( ) × ( 10
...
5 )
7

= (22) × (224
...
5 𝑚2

Therefore required area of track is 𝟒𝟗𝟑𝟑
...
Find the radius of resultant circle which has circumference equal to the sum of the circumferences of the
two circles or area equal to the sum of areas of two circles
...

Step 2: Find the circumference/area of resultant circle:
Case 1: Find the sum of circumference of two given circles to find the circumference of the resultant circle, if
circumference is asked
...

Step 3: Find the radius of the required circle:
Case 1: Equate the circumference of the circle with the circumference of the circle formula to find the radius of
resultant
...


Example 1:
The radii of two circles are 19 cm and 9 cm respectively
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Solution:
Step 1: Find the circumference\area of two circles using given radius:
Let 𝑅 and 𝑟 be the radii of two given circles and radius of resultant circle 𝑅𝑟
Given, 𝑅 = 19 cm, 𝑟 = 9 cm and
Now, circumference of circle with radius 𝑅 = 2𝜋𝑅
Circumference 𝐶1 = 2𝜋(19)
= 38 𝜋 cm
and
Circumference of circle with radius 𝑟 = 2𝜋𝑟
Circumference 𝐶2 = 2𝜋 (9)
= 18 𝜋
= 18 𝜋 cm

Step 2: Find the circumference/area of resultant circle:
Resultant circumference of two circles = 𝐶1 + 𝐶2
= 38 𝜋 + 18 𝜋
= 56 𝜋
Resultant circumference of two circles = 56 𝜋 cm
Step 3: Find the radius of the required circle:
Circumference of resultant circle = Resultant circumference of two circles =
= 2𝜋𝑅𝑟
56 𝜋 = 2𝜋𝑅𝑟
56 = 2𝑅𝑟
Radius of the circle 𝑅𝑟 =

56
2

= 28 cm
...


Example 2:
The radii of two circles are 8 cm and 6 cm respectively
...


Solution:
Step 1: Find the circumference\area of two circles using given radius:
Let 𝑅 and 𝑟 be the radii of two given circles and radius of resultant circle 𝑅𝑟
Given, 𝑅 = 8 cm, 𝑟 = 6 cm and
Now
Area of circle with radius 𝑅 = 𝜋𝑅2
= 𝜋 (8)2
Thus, area of circle 𝐴1 = 64 𝜋𝑐𝑚2
and
Area of circle with radius 𝑅 = 𝜋𝑅2
= 𝜋 (6)2
Area of circle 𝐴2 = 36 𝜋𝑐𝑚2

Step 2: Find the circumference/area of resultant circle:
Resultant Area of two circles = 𝐴1 + 𝐴2
= 64 𝜋 + 36 𝜋
= 100 𝜋
Resultant Area of two circles is 100 𝜋 cm2
Step 3: Find the radius of the required circle:
Area of circle = 𝜋𝑅𝑟2 = Resultant Area of two circles
100 𝜋 = 𝜋𝑅𝑟2
100 = 𝑅𝑟2
Radius of the circle 𝑅𝑟 = 10
Thus, radius of the circle 𝑹 = 𝟏𝟎 cm
...

Steps required to follow:
Step 1: Find the area of the minor sector:
Substitute the values of angle of sector (𝜃) and radius (𝑟) in the area of sector formula

𝜃
360

× 𝜋𝑟 2 and simplify it
...

Or
Substitute the values of angle of minor sector(𝜃) and radius(𝑟) in area of major sector formula
360−𝜃
360

× 𝜋𝑟 2 to find the area of a major sector
...


Solution:
Step 1: Find the area of the minor sector:
Given, Radius of circle (𝑟) = 14 cm
Angle of sector (𝜃) = 60
Now, Area of sector =
=

60
360

× 𝜋(14)2

𝜃
360

× 𝜋𝑟 2

=

1
6

× 𝜋(196)
22

= (98) ( ) (3)
7

Therefore, area of minor sector is 𝟏𝟎𝟐
...


Solution:
Step 1: Find the area of major sector:
Given, Radius of circle (𝑟) = 21 cm
Angle of minor sector (𝜃) = 120
Now
Area of major sector =
=
=

240
360
2
3

360−𝜃
360

× 𝜋𝑟 2

× 𝜋(21)2

× 𝜋(441)

= 2𝜋(147)
Therefore, area of major sector is 𝟗𝟐𝟑
...

Case 1: Substitute the given angle of sector in the area of sector formula and equate it to the given area of sector to
find radius of sector
...


Question 1:
The area of a sector 𝑂 − 𝐴𝐵𝐶 is equal to 125 𝜋 𝑐𝑚2 and radius of a sector is 15 cm
...


Solution:

Step 1: Find the angle of sector
...


Question 2:
The area of a sector 𝑂 − 𝐴𝐵𝐶 is equal to 250𝜋 𝑐𝑚2 and angle of a sector is 200°
...


Solution:
Step 1: Find the radius of sector
...

Given, Angle of sector = 200°
Area of sector = 250𝜋 𝑐𝑚2
Now
Area of sector =
250𝜋 =
250 =

𝟐𝟎𝟎
360
𝟓
9

𝜃
360

× 𝜋𝑟 2

× 𝜋𝑟 2

× 𝑟2

250 × 9 = 5𝑟 2
2250 = 5𝑟 2
𝑟 2 = 450
𝑟 = 21
...
𝟐𝟏 cm
...

Steps required to follow:
Step 1: Find the length of sector or radius of circle:
Case 1: Substitute the values of angle of sector and radius in the length of arc formula

𝜃
360

× 2𝜋r and simplify it to find

the length of the Arc
...


Question 1:
The angle described by sector 0 − 𝐴𝐵𝐶 is 45 degree and radius of sector is 14 cm , then find the value of length of arc

Solution:
Step 1: Find the length of sector or radius of circle:
Given,
Angle of sector 𝜃 = 45
And radius 𝑟 = 14 cm
We have, Length of arc =
=
=

45
360
1
8

𝜃
360

× 2𝜋r

× 2𝜋14

× 28 ×

22
7

= 11 cm
Therefore, the length of arc is 𝟏𝟏 cm
...


Solution:
Step 1: Find the radius of circle:
Given, angle of sector (𝜃) = 135
And the length of arc = 28 cm
We have, Length of arc =

𝜃
360

× 2𝜋r

28 =

135
360

× 2𝜋r

3

28 = × 2𝜋r
8

Radius( 𝑟 ) = 11
...
𝟖𝟕 cm
...

Steps required to follow:
Step 1: Find the area of sector:
Substitute the radius and angle of sector in the area of corresponding sector using formula

𝜃
360

× 𝜋𝑟 2
...

Step 3: Find the area of the segment:
Subtract area of corresponding triangle from area of sector to find the area of segment
...

Find (i) the length of the arc, (ii) the area of the sector, (iii) the area of the minor segment
...

Given, radius of circle (𝑟) = 21 cm
and angle of sector (𝜃) = 60°
We know that, (i) Length of the arc 𝐴𝐶𝐵 = 2𝜋𝑟(

𝜃
360

)

= (2 ×

22
7

× 21 ×

60
360

) cm

= 22cm
...

Step 2: Find the area of the corresponding triangle 𝑶𝑨𝑩:
We have, Area of triangle 𝑂𝐴𝐵 =
Area of triangle 𝑂𝐴𝐵 =
=

1
2

1
2

1
2

𝑟 2 sin 𝜃

(21)2 sin 60

(441) sin 60

= 190
...
𝟕𝟑 𝒄𝒎𝟐
Step 3: Find the area of the segment
...
73
= 40
...


Question 2:
In a circle of radius 14 cm, an arc subtends an angle of 120 at the center
...


Solution:
Step 1: Find the area of sector
Let 𝐴𝐶𝐵 be the given arc subtending an angle of 120° at the center
...
34 𝑐𝑚2
...

Step 2: Find the area of the corresponding triangle 𝑶𝑨𝑩:
1

Area of triangle 𝑂𝐴𝐵 = 𝑟 2 sin 𝜃
2
1

Area of triangle 𝑂𝐴𝐵 = (14)2 sin 120
2

1

= (196) sin 120
2

= 49(3
...
86 𝑐𝑚2
Therefore , the area of triangle = 153
...

Area of segment = area of sector 𝐴𝑂𝐵𝐴 − area of triangle 𝑂𝐴𝐵
= 205
...
86
= 51
...
𝟒𝟖 𝒄𝒎𝟐
Template 9: Find the area of the quadrant of a circle whose circumference or radius are given
...

Step 2: Find the area of quadrant:
Substitute the radii in the area of the circle formula and find the area of the circle
...


Question 1:
Find the area on the quadrant for a circle whose circumference is 198 cm
...

Circumference of circle = 2𝜋𝑟
= 198
Therefore, radius 𝑟 = (

198 × 7
2 × 22

)

𝑟 = 15
...
5 cm
Step 2: Find the area of quadrant:
Area of circle = 𝜋𝑟 2
= (3
...
5)2
= 754
...
38 𝑐𝑚2
Now
Required area of quadrant =
=

𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
4

754
...
6 𝑐𝑚2
Hence, area of quadrant is 𝟏𝟖𝟖
...
then find the area of the quadrant for the given
circle
...
14)(14)2
= 616
Thus, Area of circle = 616 𝑐𝑚2

Now , required area of quadrant =
=

𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
4

616
4

= 154 𝑐𝑚2
Hence, area of quadrant is 𝟏𝟓𝟒 𝒄𝒎𝟐
...

Step 2: Find the area of combined plane figure:
According to the given problem, find the area of the shaded region in the combined figure by
adding or subtracting corresponding plain figures
...
If asked to find the cost of painting etc
...

Find the area of the shaded region
...

Let these regions meet at a common point 𝑂
...
14 × 3
...
25)
= 100 − 78
...
5 𝑐𝑚2
Similarly, (area of II) + (area of IV) = 21
...

Step 2: Find the area of combined plane figure:
Area of shaded region = 𝑎𝑟(𝑠𝑞 𝐴𝐵𝐶𝐷) – 𝑎𝑟(𝐼 + 𝐼𝐼 + 𝐼𝐼𝐼 + 𝐼𝑉)
= 100 − (21
...
5 )
= 100 − 43
Thus, area of shaded region is 𝟓𝟕 𝒄𝒎^𝟐
...
If the center of each circular flower bed is the point of intersection 𝑂 of the diagonals of the square lawn, find (i)
the sum of the areas of the lawn and the flower beds, (ii) the sum of the areas of two flower beds
...

So, 𝐴𝐶 = 𝐵𝐷
Diagonal of square (𝐴𝐶) = √2 × 𝐴𝐵
= √2 × 56 = 56 √2
So, 𝑂𝐴 = 𝑂𝐵 =

1
2

1

𝐴𝐶 = 56 √2 = 28√2𝑚
2

Step 2: Find the area of combined plane figure:
Let 𝑂𝐴 = 𝑂𝐵 = 𝑟 𝑚 [radius of the sector]
Area of sector 𝑂𝐴𝐵 = [
1

1

22

4

4

7

= ( ) 𝜋𝑟 2 = ×
1

22

4

7

=[ ×

90°
360°

] 𝜋𝑟 2

28 2

× ( ) 𝑚2
√2

× 28 × 28 × 2] 𝑚2

= 1232 𝑚2
Area of flower bed 𝐴𝐵 = area of sector 𝑂𝐴𝐵 − area of Δ𝑂𝐴𝐵
1

1

2

2

⇒ 1232 − × 𝑂𝐵 × 𝑂𝐴 ⇒ 1232 − × 28√2 × 28√2
⇒ 1232 − 784 = 448 𝑚2
Similarly, area of square 𝐴𝐵𝐶𝐷 + area of flower bed 𝐴𝐵 + area of flower bed 𝐶𝐷
= (56 × 56) + 448 + 448 = 4032 𝑐𝑚2

Template 11: Find the area of the remaining portion after removing certain shapes from a given shape
...

Step 2: Find the area of remaining plane figure:
According to the given problem, find the area of the given figure and subtract the area of removed
portions from it to find the area of the remaining figure
...


If asked to find the cost of painting etc
...
Find the area of the shaded region
...

As 𝐴𝐵𝐶 is a quadrant of the circle, ∠𝐵𝐴𝐶 will be measured 90°
...
So substitute radius of sector
and perimeter of sector and simplify it to find length of arc
...


Question 1:
The perimeter of a sector of a circle of radius 14 cm is 68 cm
...


Solution:
Step 1: Find the length of arc:
Let 𝑂 be the center of a circle of radius 14 cm
...

Then, 𝑂𝐴 + 𝑂𝐵 + 𝑎𝑟𝑐(𝐴𝐶𝐵) = 68 cm
14 𝑐𝑚 + 14 𝑐𝑚 + 𝑎𝑟𝑐(𝐴𝐶𝐵) = 68
𝑎𝑟𝑐(𝐴𝐶𝐵) = 68 − 14 − 14
∴ 𝑎𝑟𝑐(𝐴𝐶𝐵) = 40 cm
Step 2: Find the area of sector:
1

∴ 𝑎𝑟(𝑠𝑒𝑐𝑡𝑜𝑟 𝑂𝐴𝐶𝐵𝑂) = ( × 𝑟𝑎𝑑𝑖𝑢𝑠 × 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ )
2

= ( × 14 × 40 )
= 280 𝑐𝑚2

∴ 𝑎𝑟(𝑠𝑒𝑐𝑡𝑜𝑟 𝑂𝐴𝐶𝐵𝑂) = 280 𝑐𝑚2

Example 2:
In a circle of radius 21 cm, an arc subtends an angle of 60 at the center
...

Given radius of circle (𝑟) = 21 cm
and angle of minor sector (𝜃) = 60°
(i) Length of the arc 𝐴𝐶𝐵 = 2𝜋r (
= (2 ×

22
7

× 21 ×

𝜃
360

)

60
360

) cm

= 22cm
...

∴ Area of the sector 𝐴𝐶𝐵𝑂𝐴 = 231 𝑐𝑚2
...
e
...
Find the angle made by a minute hand

in the given time
...


Question 1:
The length of the minute hand of a clock is 14cm
...


Solution:
Step 1: Find the angle made by minute hand for the given time:
Given, length of the minute hand = 14 cm
...
𝟐𝟖𝟔 𝒄𝒎𝟐

Question 2:
The length of the minute hand of a clock is 10cm
...
25am
...
25am= 30𝑜
Step 2: Substitute angle and radius to find required area:

Area of face of clock described = Area of sector = 𝜋𝑟 2 (
=
=

30
360
1
12

𝜃
360

)

× 𝜋 × 102

× 100𝜋

Thus, required area of sector is 𝟐𝟔
...


•

Then find the distance covered by the wheel in given time
...
As the
circumference of a circular shaped wheel is nothing but the distance travelled by car in one
complete revolution
...


Question 1:
A car has wheels which are 80 cm in diameter
...


Question 2:
The wheel of a motor cycle is of radius 35 cm
...
35 cm
The speed it must keep = 𝑠 = 66 km/Hence, the answer is =

66 ×100
60

m/min = 1100 m/min

Step 2: Find the circumference of the wheel:
The circumference = 𝐶 of the wheel = 2𝜋𝑟 = 2 ×

22
7

× 035 𝑚 = 2
...

Now the distance = 𝑑 covered by a wheel in one revolution = the circumference of the wheel
...


=

1100
2

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