Surface Area and Volume of a Right Circular Cone- 2 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Q u e s t i o n : 2 4
Find the volume of a right circular cone with:
i
radius 6 cm, height 7 cm.
ii
radius 3.5 cm, height 12 cm
iii
height 21 cm and slant height 28 cm.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i Substituting the values of r = 6 cm and h = 7 cm in the above equation and using 
Volume = 
=
22
2
6
= 264
Hence the volume of the given cone with the specified dimensions is 
ii Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using 
Volume = 
=
22
0.5
3.5
4
= 154
Hence the volume of the given cone with the specified dimensions is 
iii In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm.
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.
Page 2


Q u e s t i o n : 2 4
Find the volume of a right circular cone with:
i
radius 6 cm, height 7 cm.
ii
radius 3.5 cm, height 12 cm
iii
height 21 cm and slant height 28 cm.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i Substituting the values of r = 6 cm and h = 7 cm in the above equation and using 
Volume = 
=
22
2
6
= 264
Hence the volume of the given cone with the specified dimensions is 
ii Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using 
Volume = 
=
22
0.5
3.5
4
= 154
Hence the volume of the given cone with the specified dimensions is 
iii In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm.
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.
Substituting the values of r =  cm and h = 21 cm in the above equation and using 
Volume = 
=
22
343
= 7546
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 5
Find the capacity in litres of a conical vessel with:
i
radius 7 cm, slant height 25 cm
ii
height 12 cm, slant height 13 cm.
 
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i In a cone, the base radius ‘r’ is given as 7 cm and the slant height ‘l’ is given as 25 cm.
To find the base vertical height ‘h’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 24
Therefore the vertical height is, h = 24 cm.
Substituting the values of r = 7 cm and h = 24 cm in the above equation and using 
Volume = 
=
22
7
8
= 1232
Hence the volume of the given cone with the specified dimensions is 
ii In a cone, the vertical height ‘h’ is given as 12 cm and the slant height ‘l’ is given as 13 cm.
Page 3


Q u e s t i o n : 2 4
Find the volume of a right circular cone with:
i
radius 6 cm, height 7 cm.
ii
radius 3.5 cm, height 12 cm
iii
height 21 cm and slant height 28 cm.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i Substituting the values of r = 6 cm and h = 7 cm in the above equation and using 
Volume = 
=
22
2
6
= 264
Hence the volume of the given cone with the specified dimensions is 
ii Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using 
Volume = 
=
22
0.5
3.5
4
= 154
Hence the volume of the given cone with the specified dimensions is 
iii In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm.
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.
Substituting the values of r =  cm and h = 21 cm in the above equation and using 
Volume = 
=
22
343
= 7546
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 5
Find the capacity in litres of a conical vessel with:
i
radius 7 cm, slant height 25 cm
ii
height 12 cm, slant height 13 cm.
 
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i In a cone, the base radius ‘r’ is given as 7 cm and the slant height ‘l’ is given as 25 cm.
To find the base vertical height ‘h’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 24
Therefore the vertical height is, h = 24 cm.
Substituting the values of r = 7 cm and h = 24 cm in the above equation and using 
Volume = 
=
22
7
8
= 1232
Hence the volume of the given cone with the specified dimensions is 
ii In a cone, the vertical height ‘h’ is given as 12 cm and the slant height ‘l’ is given as 13 cm.
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 5
Therefore the base radius is, r = 5 cm.
Substituting the values of r = 5 cm and h = 12 cm in the above equation and using 
Volume = 
= 314.28
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 6
Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the heights of the two cones is 1: 3.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 1k
= 3k
It is also given that the ratio between the base radiuses of the two cones is 3: 1.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 3p
= 1p
Substituting these values in the formula for volume of cone we get,
= 
= 
Hence we see that the ratio between the volumes of the two given cones is 
Q u e s t i o n : 2 7
Page 4


Q u e s t i o n : 2 4
Find the volume of a right circular cone with:
i
radius 6 cm, height 7 cm.
ii
radius 3.5 cm, height 12 cm
iii
height 21 cm and slant height 28 cm.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i Substituting the values of r = 6 cm and h = 7 cm in the above equation and using 
Volume = 
=
22
2
6
= 264
Hence the volume of the given cone with the specified dimensions is 
ii Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using 
Volume = 
=
22
0.5
3.5
4
= 154
Hence the volume of the given cone with the specified dimensions is 
iii In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm.
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.
Substituting the values of r =  cm and h = 21 cm in the above equation and using 
Volume = 
=
22
343
= 7546
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 5
Find the capacity in litres of a conical vessel with:
i
radius 7 cm, slant height 25 cm
ii
height 12 cm, slant height 13 cm.
 
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i In a cone, the base radius ‘r’ is given as 7 cm and the slant height ‘l’ is given as 25 cm.
To find the base vertical height ‘h’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 24
Therefore the vertical height is, h = 24 cm.
Substituting the values of r = 7 cm and h = 24 cm in the above equation and using 
Volume = 
=
22
7
8
= 1232
Hence the volume of the given cone with the specified dimensions is 
ii In a cone, the vertical height ‘h’ is given as 12 cm and the slant height ‘l’ is given as 13 cm.
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 5
Therefore the base radius is, r = 5 cm.
Substituting the values of r = 5 cm and h = 12 cm in the above equation and using 
Volume = 
= 314.28
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 6
Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the heights of the two cones is 1: 3.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 1k
= 3k
It is also given that the ratio between the base radiuses of the two cones is 3: 1.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 3p
= 1p
Substituting these values in the formula for volume of cone we get,
= 
= 
Hence we see that the ratio between the volumes of the two given cones is 
Q u e s t i o n : 2 7
The radius and the height of a right circular cone are in the ratio 5 : 12. If its volume is 314 cubic metre, find the slant
height and the radius (Use p = 3. 14
).
S o l u t i o n :
It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.
Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.
So, 
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
The volume of the cone is given as 
Substituting the values of  and  and using  in the formula for the volume of a cone,
Volume = 
314 = 
= 1
k = 1
Therefore the actual value of the base radius is r = 5 m and h = 12 m.
Hence the radius of the cone is 
We are given that r = 5 m and h = 12 m. We find l using the relation
= 
= 
= 
= 13.
Therefore, the slant height of the given cone is 
Hence the radius of cone and slant height is 5 m and 13 m respectively
Q u e s t i o n : 2 8
The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the slant height
and radius of the cone.
(Use p = 3. 14
).
S o l u t i o n :
It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.
Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.
So, 
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Page 5


Q u e s t i o n : 2 4
Find the volume of a right circular cone with:
i
radius 6 cm, height 7 cm.
ii
radius 3.5 cm, height 12 cm
iii
height 21 cm and slant height 28 cm.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i Substituting the values of r = 6 cm and h = 7 cm in the above equation and using 
Volume = 
=
22
2
6
= 264
Hence the volume of the given cone with the specified dimensions is 
ii Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using 
Volume = 
=
22
0.5
3.5
4
= 154
Hence the volume of the given cone with the specified dimensions is 
iii In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 28 cm.
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.
Substituting the values of r =  cm and h = 21 cm in the above equation and using 
Volume = 
=
22
343
= 7546
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 5
Find the capacity in litres of a conical vessel with:
i
radius 7 cm, slant height 25 cm
ii
height 12 cm, slant height 13 cm.
 
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
i In a cone, the base radius ‘r’ is given as 7 cm and the slant height ‘l’ is given as 25 cm.
To find the base vertical height ‘h’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 24
Therefore the vertical height is, h = 24 cm.
Substituting the values of r = 7 cm and h = 24 cm in the above equation and using 
Volume = 
=
22
7
8
= 1232
Hence the volume of the given cone with the specified dimensions is 
ii In a cone, the vertical height ‘h’ is given as 12 cm and the slant height ‘l’ is given as 13 cm.
To find the base radius ‘r’ we use the relation between r, l and h.
We know that in a cone
= 
= 
= 
= 5
Therefore the base radius is, r = 5 cm.
Substituting the values of r = 5 cm and h = 12 cm in the above equation and using 
Volume = 
= 314.28
Hence the volume of the given cone with the specified dimensions is 
Q u e s t i o n : 2 6
Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the heights of the two cones is 1: 3.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 1k
= 3k
It is also given that the ratio between the base radiuses of the two cones is 3: 1.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 3p
= 1p
Substituting these values in the formula for volume of cone we get,
= 
= 
Hence we see that the ratio between the volumes of the two given cones is 
Q u e s t i o n : 2 7
The radius and the height of a right circular cone are in the ratio 5 : 12. If its volume is 314 cubic metre, find the slant
height and the radius (Use p = 3. 14
).
S o l u t i o n :
It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.
Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.
So, 
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
The volume of the cone is given as 
Substituting the values of  and  and using  in the formula for the volume of a cone,
Volume = 
314 = 
= 1
k = 1
Therefore the actual value of the base radius is r = 5 m and h = 12 m.
Hence the radius of the cone is 
We are given that r = 5 m and h = 12 m. We find l using the relation
= 
= 
= 
= 13.
Therefore, the slant height of the given cone is 
Hence the radius of cone and slant height is 5 m and 13 m respectively
Q u e s t i o n : 2 8
The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the slant height
and radius of the cone.
(Use p = 3. 14
).
S o l u t i o n :
It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.
Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.
So, 
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
The volume of the cone is given as 
Substituting the values of  and  and using in the formula for the volume of a cone,
Volume = 
2512 = 
= 8
k = 2
Therefore the actual value of the base radius is r = 10 cm and h = 24 cm.
Hence the radius of the cone is 
We are given that r = 10 cm and h = 24 cm. We find l using the relation
= 
= 
= 
= 26
Therefore the slant height of the given cone is 
Hence the radius and slant height of the cone are 10 cm and 26 cm respectively
Q u e s t i o n : 2 9
The ratio of volumes of two cones is 4 : 5 and the ratio of the radii of their bases is 2 : 3, Find the ratio of their vertical
heights.
S o l u t i o n :
The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as
Volume = 
Let the volume, base radius and the height of the two cones be  and  respectively.
It is given that the ratio between the volumes of the two cones is 4: 5.
Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.
So,  = 4k
= 5k
It is also given that the ratio between the base radiuses of the two cones is 2: 3.
Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.
So,  = 2p
= 3p
Substituting these values in the formula for volume of cone we get,
=