Surface Area and Volume of a Right Circular Cone- 1 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Q u e s t i o n : 1
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Substituting the values of r = 21 cm and l = 60 cm in the above equation and using 
Curved Surface Area will be,
= 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 2
The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 5 cm and vertical height h = 12 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l= 
= 
= 
= 
l = 13 cm
Now, substituting the values of r = 5 cm and slant height l = 13 cm and using  in the formula of C.S.A,
We get Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 3
The radius of a cone is 7 cm and area of curved surface is 176 cm
2
. Find the slant height.
S o l u t i o n :
Page 2


Q u e s t i o n : 1
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Substituting the values of r = 21 cm and l = 60 cm in the above equation and using 
Curved Surface Area will be,
= 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 2
The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 5 cm and vertical height h = 12 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l= 
= 
= 
= 
l = 13 cm
Now, substituting the values of r = 5 cm and slant height l = 13 cm and using  in the formula of C.S.A,
We get Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 3
The radius of a cone is 7 cm and area of curved surface is 176 cm
2
. Find the slant height.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 176 cm
2
 and that the base radius is 7 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 8
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 4
The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.
S o l u t i o n :
In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm, and the area of the base is
asked. The base area is given as
Base area = 
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
Therefore the base radius is, r =  cm.
Now, let us substitute the value of r in the formula for area of the base.
Base Area = 
= 
= 
= 1078
Hence, the base area of the cone with the specified dimensions is .
Q u e s t i o n : 5
Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
Page 3


Q u e s t i o n : 1
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Substituting the values of r = 21 cm and l = 60 cm in the above equation and using 
Curved Surface Area will be,
= 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 2
The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 5 cm and vertical height h = 12 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l= 
= 
= 
= 
l = 13 cm
Now, substituting the values of r = 5 cm and slant height l = 13 cm and using  in the formula of C.S.A,
We get Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 3
The radius of a cone is 7 cm and area of curved surface is 176 cm
2
. Find the slant height.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 176 cm
2
 and that the base radius is 7 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 8
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 4
The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.
S o l u t i o n :
In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm, and the area of the base is
asked. The base area is given as
Base area = 
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
Therefore the base radius is, r =  cm.
Now, let us substitute the value of r in the formula for area of the base.
Base Area = 
= 
= 
= 1078
Hence, the base area of the cone with the specified dimensions is .
Q u e s t i o n : 5
Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
But we do not have the slant height. We are given that r = 6 cm and h = 8 cm. We find l using the relation
= 
= 
= 
= 10.
Therefore, the slant height, l = 10 cm.
Substituting the values of r = 6 cm and l = 10 cm in the above equation and using  in specified formula,
Total Surface Area = 
= 
= 
Therefore the total surface area of the given cone is or 301.71 cm
2
.
Q u e s t i o n : 6
Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Substituting the values of r = 5.25 cm and l = 10 cm in the above equation and using 
Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is 
Q u e s t i o n : 7
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
The diameter of the base is given as 24 m. The radius of the base is half of the diameter and hence r = 12 m.
Substituting the values of r = 12 m and l = 21 cm in the above equation and using  in specified formula,
Total Surface Area = 
Page 4


Q u e s t i o n : 1
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Substituting the values of r = 21 cm and l = 60 cm in the above equation and using 
Curved Surface Area will be,
= 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 2
The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 5 cm and vertical height h = 12 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l= 
= 
= 
= 
l = 13 cm
Now, substituting the values of r = 5 cm and slant height l = 13 cm and using  in the formula of C.S.A,
We get Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 3
The radius of a cone is 7 cm and area of curved surface is 176 cm
2
. Find the slant height.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 176 cm
2
 and that the base radius is 7 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 8
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 4
The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.
S o l u t i o n :
In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm, and the area of the base is
asked. The base area is given as
Base area = 
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
Therefore the base radius is, r =  cm.
Now, let us substitute the value of r in the formula for area of the base.
Base Area = 
= 
= 
= 1078
Hence, the base area of the cone with the specified dimensions is .
Q u e s t i o n : 5
Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
But we do not have the slant height. We are given that r = 6 cm and h = 8 cm. We find l using the relation
= 
= 
= 
= 10.
Therefore, the slant height, l = 10 cm.
Substituting the values of r = 6 cm and l = 10 cm in the above equation and using  in specified formula,
Total Surface Area = 
= 
= 
Therefore the total surface area of the given cone is or 301.71 cm
2
.
Q u e s t i o n : 6
Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Substituting the values of r = 5.25 cm and l = 10 cm in the above equation and using 
Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is 
Q u e s t i o n : 7
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
The diameter of the base is given as 24 m. The radius of the base is half of the diameter and hence r = 12 m.
Substituting the values of r = 12 m and l = 21 cm in the above equation and using  in specified formula,
Total Surface Area = 
= 
= 
Therefore the total surface area of the given cone is or 1244.57 m
2
.
Q u e s t i o n : 8
The area of the curved surface of a cone is 60 p
cm
2
. If the slant height of the cone be 8 cm, find the radius of the base.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is  cm
2 
and that the slant height is 8 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, r = 
Substituting the values of C.S.A and the slant height in the above equation,
Slant height, r = 
= 7.5
Hence the base radius of the cone with the mentioned dimensions is .
Q u e s t i o n : 9
The curved surface area of a cone is 4070 cm
2
 and its diameter is 70 cm. What is its slant height? (Use p
= 22/7).
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 4070 cm
2
 and that the base diameter is 70 cm. The formula of the curved surface area of a cone with
base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
The base radius is half of the base diameter. And since the base diameter is given as 70 cm we can find out the base
radius as, r = 35 cm.
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 
= 37
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 1 0
Page 5


Q u e s t i o n : 1
Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Substituting the values of r = 21 cm and l = 60 cm in the above equation and using 
Curved Surface Area will be,
= 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 2
The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 5 cm and vertical height h = 12 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l= 
= 
= 
= 
l = 13 cm
Now, substituting the values of r = 5 cm and slant height l = 13 cm and using  in the formula of C.S.A,
We get Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is .
Q u e s t i o n : 3
The radius of a cone is 7 cm and area of curved surface is 176 cm
2
. Find the slant height.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 176 cm
2
 and that the base radius is 7 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 8
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 4
The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.
S o l u t i o n :
In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm, and the area of the base is
asked. The base area is given as
Base area = 
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
Therefore the base radius is, r =  cm.
Now, let us substitute the value of r in the formula for area of the base.
Base Area = 
= 
= 
= 1078
Hence, the base area of the cone with the specified dimensions is .
Q u e s t i o n : 5
Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
But we do not have the slant height. We are given that r = 6 cm and h = 8 cm. We find l using the relation
= 
= 
= 
= 10.
Therefore, the slant height, l = 10 cm.
Substituting the values of r = 6 cm and l = 10 cm in the above equation and using  in specified formula,
Total Surface Area = 
= 
= 
Therefore the total surface area of the given cone is or 301.71 cm
2
.
Q u e s t i o n : 6
Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.
S o l u t i o n :
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = prl
Substituting the values of r = 5.25 cm and l = 10 cm in the above equation and using 
Curved Surface Area = 
= 
= 
Therefore the Curved Surface Area of the cone with the specified dimensions is 
Q u e s t i o n : 7
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
S o l u t i o n :
The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Total Surface Area = 
The diameter of the base is given as 24 m. The radius of the base is half of the diameter and hence r = 12 m.
Substituting the values of r = 12 m and l = 21 cm in the above equation and using  in specified formula,
Total Surface Area = 
= 
= 
Therefore the total surface area of the given cone is or 1244.57 m
2
.
Q u e s t i o n : 8
The area of the curved surface of a cone is 60 p
cm
2
. If the slant height of the cone be 8 cm, find the radius of the base.
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is  cm
2 
and that the slant height is 8 cm. The formula of the curved surface area of a cone with base
radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, r = 
Substituting the values of C.S.A and the slant height in the above equation,
Slant height, r = 
= 7.5
Hence the base radius of the cone with the mentioned dimensions is .
Q u e s t i o n : 9
The curved surface area of a cone is 4070 cm
2
 and its diameter is 70 cm. What is its slant height? (Use p
= 22/7).
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 4070 cm
2
 and that the base diameter is 70 cm. The formula of the curved surface area of a cone with
base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Hence, slant height, l = 
The base radius is half of the base diameter. And since the base diameter is given as 70 cm we can find out the base
radius as, r = 35 cm.
Substituting the values of C.S.A and the base radius and using  in the above equation,
Slant height, l = 
= 
= 37
Hence the slant height of the cone with the mentioned dimensions is .
Q u e s t i o n : 1 0
The radius and slant height of a cone are in the ratio of 4 : 7. If its curved surface area is 792 cm
2
, find its radius. (Use p
= 22/7).
S o l u t i o n :
It is given that the curved surface area
C. S. A of the cone is 792 cm
2
 and that the ratio between the base radius and the slant height is 4: 7. The formula of the
curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
Since only the ratio between the base radius and the slant height is given, we shall use them by introducing a constant ‘k’
So, r = 4k
l = 7k
Substituting the values of C.S.A, base radius, slant height and using  in the above equation,
Curved Surface Area,
792 = 
792 = 
9 = 
Hence the value of k = 3.
From this we can find the value of base radius,
r = 4k
r = 12
Therefore the base radius of the cone is .
Q u e s t i o n : 1 1
A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet
required to make 10 such caps.
S o l u t i o n :
The area of sheet required to make a cone would be equal to the curved surface area of the cone that is to be formed. So
here we need to find the C.S.A. of a single cone and then multiply the same to arrive at the final answer.
The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as
Curved Surface Area = 
But, here we’re given only that the base radius r = 7 cm and vertical height h = 24 cm.
To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation
Slant height, l = 
= 
= 
= 
l = 25 cm
Now, substituting the values of r = 7 cm and slant height l = 25 cm and using  in the formula of C.S.A,
We get Curved Surface Area =