Chapter 12. Areas Related to Circles – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions
CBSE Class 10 Mathematics Chapter 12. Areas Related to Circles Important Questions with Solutions and Answers
Chapter 12. Areas Related to Circles
SECTION = A
Question: Area of a sector of angle P (in degree) of a circle with radius R is : [SEBA 2015]
(a) 
(b) 
(c) 
(d) 
Solution: (d)
Question: If the sum of the areas of two circles with radii 
and 
is equal to the area of a circle of radius,then
(a) 
(b) 
(c) 
(d) 
Solution: (b)
Question: 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
Solution: (b) 14 : 11 .
[ let 
be the radius of the circle and 
be side of the square .
A/Q , 
]
Question: If the circumference of a circle is 44 cm , then its area is : [SEBA 2016]
(a) 276 
(a) 44 (c) 176 (d) 154
Solution: (d) 154 .
[ A/Q, 
cm
The area of the circle 
]
Question: The area of the circle that can be inscribe in a square of side 6 cm is : [ SEBA 2017]
(a) 
(b) 
(c) 
(d) 
Solution: (d)
[ The square of the side 
cm and the radius 
The area of the circle 
]
Question: If the circumference of a circle is 22 cm , then the area of a quadrant of the circle is : [SEBA 2018]
(a) 
(b) 77 (c) 
(d) 
Solution: (a)
[ A/Q , 
The area of a quadrant of the circle 
]
Question: The degree measure of the angle at the centre of a circle is 1 . The area of the sector is : [SEBA 2019]
(a) 
(b) 
(c) 
(d) 
Solution: (d)
Question: The degree measure of the angle at the centre of a circle is θ . The length of a arc of the sector is : [ SEBA 2020]
(a) 
(b) 
(c) 
(d) 
Where is the radius of the circle .
Solution: (b)
Question: If the sum of the circumference of two circles with radii 
and 
is equal to the circumference of a circle of radius , then
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ A/Q, 
]
Question: If the radius of a circle is doubled , then the ratio of the areas of the new circle to the area of the given circle is :
(a) 2 : 1 (b) 1 : 2 (c) 1 : 4 (d) 4 :1
Solution: (d) 4 :
[ Here , 
and 
]
Question: In the given figure , if the angle 30° , then the area of the shaded region (in cm square) is : [2011]
(a) 3.36 (b) 2.26 (c) 1.36 (d) 2.36
Solution: (d) 2.36 [ Here , 
cm and 
cm
The area of shaded region 
]
Question: If the area of a circle is equal to the sum of the areas of two circles of diameters 10 cm and 24 cm, then diameter of the larger circle (in cm) is : [CBSE 2012]
(a) 34 (b) 26 (c) 17 (d) 14
Solution: (b) 26
[ Here, 
A/Q , 
cm
Therefore , the diameter 
cm 
cm ]
Question: If 
is taken as 
, the distance (in metres) covered by a wheel of diameter 35 cm, in one revolution is : [CBSE 2013]
(a) 2.2 (b) 1.54 (c) 9.625 (d) 96.25
Solution: (b) 1.54
[ Here , Diameter 
cm and Radius 
Therefore, the distance covered by a wheel in one revolution 
cm 
]
Question: 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 :
(a) 30 cm (b) 40 cm (c) 50 cm (d) 20 cm
Solution: (c) 50 cm .
[ Here , 
cm and 
cm
A/Q , 
cm
Therefore , the diameter 
cm 
cm ]
Question: Area of a sector to circle of radius 36 cm is 
, then the length arc of the corresponding arc of the circle is :
(a) 
(b) 
(c) 
(d) 
Solution: (a)
[ Here , 
cm
A/Q, 
Therefore, the length arc of the corresponding arc of the circle 
]
Question: If the diameter of a semi-circular protractor is 14 cm , then its perimeter is : [CBSE 2009]
(a) 11 cm (b) 21 cm (c) 22 cm (d) 44 cm
Solution: (c) 22 cm .
[ Here , 
cm and 
The perimeter of a semi-circular 
]
Answers Following the Questions :
Question: In given figure , find the area of the sector of a circle with radius 4 cm .
Solution: Here , cm and 
Therefore, the area of the sector of a circle 
Question: Find the length of the arc of a circle of diameter 42 cm which subtends angle of 60° at the circle ? [CBSE 2012]
Solution: Here, Radius 
and 
Therefore, the length of the arc of a circle 
Question: Find the area of a quadrant of a circle whose circumference is 
. [ CBSE 2014]
Solution: Let, r be the radius of a circle .
A/Q , 
Therefore , the area of quadrant of the circle 
Question: The difference between the circumference and radius of a circle is 37 cm . Find the area of the circle . [ CBSE 2013]
Solution: Let, r be the radius of a circle .
A/Q , 
Therefore, the area of circle 
.
Question: Find the area of a sector of a circle of radius 28 cm and central angle 45° .
Solution: Here , 
cm and 
Therefore, the area of a sector of a circle 
Question: The radii of radius circles are 19 cm and 9 cm respectively . Find the radius of the circle which has circumference equal to the sum of the circumference of the two circles.
Solution: let be the radius of the new circle .
Here , 
cm and 
cm
A/Q , 
cm
SECTION = B
Question: The length of the minute hand of a clock is 14 cm . Find the area swept by the minute hand in 5 minutes .
Solution : Here, 
The angle of swept in 5 minutes 
Therefore, the area swept by the minute hand in 5 minutes 
Question: Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60° .
Solution: Here , 
cm and
Therefore, the area of a sector of a circle 
Question: The cost of fencing a circular field at the rate of Rs. 24 per metre is Rs. 5280 , then find the radius of field .
Solution: let be the radius of the circular field .
The length of the fence field 
A/Q , 
Question: An umbrella has 8 ribs which are equally spaced and the umbrella to be a flat circle of radius 45 cm , find the area between the two consecutive ribs of the umbrella .
Solution: Here , 
cm
The area between the two consecutive ribs of the umbrella 
Question: If the area of a sector of a circle of radius 14 cm is 
. Find the length of the corresponding arc of the sector . [CBSE 2011]
Solution: Here , 
cm
A/Q , 
Therefore, the length of the corresponding arc of the sector 
SECTION = C
Question: A chord of a circle of radius 10 cm subtends a right angle at the centre . Find the area of the corresponding : (i) minor segment (ii) major sector . ( Use 
)
Solution : Here, 
The area of the minor segment of a circle 
(ii) Here, 
, 
Area of major sector of a circle 
Question: In a circle of radius 21 cm, 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 (iii) area of the segment formed by the corresponding chord .
Solution: Here, Radius 
cm and 
(i) the length of an arc of a sector of a circle
(ii) the area of the sector formed by the arc
(iii) Here, OA = OB = 21 cm and 
So, OAB be an equilateral triangle .
Area of triangle OAB 
The area of segment formed by chord 
Area of sector – Area of triangle
Question: A chord of a circle of radius 15 cm subtends an angle of 60° at the centre . Find the areas of the corresponding minor and major segments of the circle . (Use 
and 
) [SEBA 2017]
Solution: Here , Radius 
cm ,
Area of the minor segment = Area of the sector – Area of the triangle formed by radius and chord
Area of major segment = Area of the circle – Area of the minor segment 
Question: A chord of a circle of radius 12 cm subtends an angle of 120° at the centre . Find the area of the corresponding segment of the circle . (Use and )
Solution : Here, 
and 
The area of the corresponding segment of the circle
Question: A brooch is made with silver wire in the form of a circle with diameter 35 mm . The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 12.12 . Find :
(i) the total length of the silver wire required . (ii) the area of each sector of the brooch .
Fig. 12.12
Solution: (i) Here, diameter 
and Radius 
The circumference 
The length of the wire required to make 5 diameters 
Therefore, the total length of the silver wire required 
(ii) The angle of each brooch 
So, the area of each sector of the brooch 
SECTION = D
Question: A round table cover has six equal designs as shown in Fig. 12.14 . If the radius of the cover is 28 cm , find the cost of making the designs at the rate of Rs 0.35 per 
. (Use )
Solution: Here, 
and 
Area of a design part of the round table cover
Area of six design part of the round table cover 
Therefore, the cost of making the designs
Question: 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 (see Fig. 12.11) . Find : (i) the area of that part of the field in which the horse can graze . (ii) the increase in the grazing area if the rope were 10 m long instead of 5 m . (Use 
)
Solution : Since , ABCD is a square .
So, AB = BC = CD = AD = 15 cm
(i) Here, 
, 
Therefore, the area of the field in which the horse can graze 
(ii) Here, 
, 
and 
The increasing area of the field 
Question: In figure, a square OPQR is inscribe in a quadrant OAQB of a circle. If the radius of the circle is 
cm , find the area of the shaded region . [Use π = 3.14] [CBSE 2020 standard]
Solution: Given,
be a square .
So , 
cm
Area of square 
In
, we have 
cm
Radius 
cm
Area of the quadrant 
Therefore, the area of shaded region 
Question: In figure , ABCD is a square with side 
cm and inscribed in a circle .Find the area of the shaded region . [Use 
] [CBSE 2019]
Solution: Given 
cm and We join AC .
In
ABC , we have
Radius 
Therefore, the area of the shaded region 
Area of circle – Area of ABCD
Q4. Find the area of the shaded design in figure, where ABCD is a square of side 10 cm and semicircles are drawn with each side of the square as diameter . [ Use 
]
Solution: Here ,
cm , Radius
cm
Area of unshaded region 
part and 
part
Area of ABCD 
Area of two semicircle
Area of unshaded region 
part and 
part .
Therefore, the area of shaded design
Area of square ABCD 
Area of 
Question: Calculate the area of the designed region in figure , common between the two quadrants of circles of radius 8 cm each . [CBSE 2012]
Solution: The radius of the quadrant of a circle 
cm and the side of the square cm
Therefore, the area of the shaded region Area of two quadrant of the circle – Area of the square
Question: In figure , a square OABC is inscribed in a quadrant OPBQ .If OA = 15 cm , find the area of the shaded region . [ Use ] [CBSE 2019]
Solution: Since OABC is a square .
So ,
cm and we join OB .
In OAB we have ,
cm
Radius
cm
Therefore , the area of shaded region
Area of quadrant of the circle – Area of square OABC
SECTION = E
Question: In figure, AB and CD are two diameter of a circle (with centre) perpendicular to each other and OD is the diameter of the smallest circle . If 
, find the area of the shaded region . [CBSE 2013,2020 Basic]
Solution: For big circle : Radius 
cm ,
Diameter 
cm
So, 
cm
For small circle : Diameter 
cm
and Radius 
cm
Area of small circle
Area of 
Area of semicircle 
The area of the shaded region
Area of small circle 
Area of semicircle – 
Question: In figure, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter . Find the area of the shaded region . [CBSE 2014]
Solution: For ABCD quadrant :
Here , Radius 
cm
In∆ABC
, we have
cm
area of 
and Area of quadrant ABCD 
For semi-circle BEC :
Diameter 
cm ; Radius 
cm
Area of semi-circle BEC 
Area of shaded region ar() ar(semicircle BEC) – ar(quadrant ABCD)
Question: In figure, ABC is a right angled triangle in which 
, 
cm and 
cm , then the semi-circles are described on AB , BC and AC as diameters . Find the area of the shaded region .
Solution: Here , cm and cm
In ABC , we have
cm
The radius of three semicircle are : 
cm ; 
cm and 
cm
Therefore, the area of the shaded region
Area of semicircle with diameter AB Area of triangle ABC Area of semicircle with diameter AC Area of semicircle with diameter BC
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