Lesson: Surface Areas and Volumes

Exercise 13.1

Write the correct answer in each of the following:

Question: 1

The radius of a sphere is 2r, and then its volume will

be:

(a)

3

4

3

(b)

3

4 r

(c)

3

8

3

r

(d)

3

32

3

r

Solution:

d

Volume of a sphere of radius

3

4

3

rr

If the radius is 2r then the volume will be

33

4 32

(2 ) .

33

rr

Question: 2

The total surface area of a cube is 96 cm

2

. The volume

of the cube is:

(a) 8 cm

3

(b) 512 cm

3

(c) 64 cm

3

(d) 27 cm

3

Solution:

c

Surface area of a cube of side

2

6aa

.

Therefore,

cm

22

6 96a

cm4a

Volume of a cube of side

3

aa

For,

4a

, the volume

3

4 64

cm

3

.

Question: 3

A cone is 8.4 cm high and the radius of its base is 2.1

cm. It is melted and recast into a sphere. The radius of

the sphere is:

(a) 4.2 cm

(b) 2.1 cm

(c) 2.4 cm

(d) 1.6 cm

Solution:

b

Volume of a cone of height h and the radius of its base

2

1

.

3

r r h

So,

The volume of the cone of height 8.4 cm and the

radius of its base

cm =

2

1

2.1 (2.1) 8.4

3

As,

The cone is melted and recast into a sphere.

Therefore,

The volume of the cone is equal to the volume of the

sphere.

Now,

Volume of a sphere of radius

3

4

.

3

rr

Then,

32

41

(2.1) 8.4

33

πr π

32

(2.1) 2.1r

2. c1 mr

Question: 4

In a cylinder, radius is doubled and height is halved,

curved surface area will be

(a) Halved

(b) Doubled

(c) Same

(d) Four times

Solution:

c

Curved surface area of a cylinder of radius r and

height

2h rh

In a cylinder, radius is doubled and height is halved,

So,

New curved surface area will be

22()

2

2.

π r h

πrh

Question: 5

The total surface area of a cone whose radius is

2

r

and

slant height 2l is

(a)

2πr l r

(b)

4

r

πr l

(c)

()πr l r

(d)

2πrl

Solution:

b

Total surface area of a right circular cone of radius r

and slant height

2

, . ., .l πrl πr i e πr l r

The total surface area of a cone whose radius is

2

r

and

slant height 2l is

()

2

22

π r r

l

4

r

πr l

Question: 6

The radii of two cylinders are in the ratio of 2:3 and

their heights are in the ratio of 5:3. The ratio of their

volumes is:

(a) 10:17

(b) 20:27

(c) 17:27

(d) 20:37

Solution:

b

Volume of a cylinder of radius r and height

2

πh r h

Therefore,

If the radii of two cylinders are in the ratio of 2:3 and

their heights are in the ratio of 5:3, the ratio of their

volumes is 20:27.

Question: 7

The lateral surface area of a cube is 256 m

2

. The

volume of the cube is

(a) 512 m

3

(b) 64 m

3

(c) 216 m

3

(d) 256 m

3

Solution:

a

The lateral surface area of a cube is 256 m

2

.

Let the side of a cube is a m.

So,

2

4 256a

m8a

Thus,

The volume of the cube is

m

33

8

i.e.

m

3

512

.

Question: 8

The number of planks of dimensions

(

m cm cm4 50 20

) that can be stored in

a pit which is 16 m long, 12 m wide and 4 m deep is

(a) 1900

(b) 1920

(c) 1800

(d) 1840

Solution:

b

Volume of the plank

m m m4 0.50 0.20

m

3

0.4

Capacity of the pit

m m m16 12 4

m

3

768

Therefore,

Number of planks

768

0.4

1920

Question: 9

The length of the longest pole that can be put in a

room of dimensions (

m m m10 10 5

) is

(a) 15 m

(b) 16 m

(c) 10 m

(d) 12 m

Solution:

a

The length of the longest pole that can be put in a

room of dimensions

m m m10 10 5

will be equal to

the diagonal of the room.

So,

The length of the longest pole is

2 2 2

10 10 5

m i.e.

15 m.

Question: 10

The radius of a hemispherical balloon increases from 6

cm to 12 cm as air is being pumped into it. The ratios

of the surface areas of the balloon in the two cases is

(a) 1:4

(b) 1:3

(c) 2:3

(d) 2:1

Solution:

a

Case I:

When the radius of the hemispherical balloon is 6 cm,

The surface area of the balloon is

2

3 (6)

i.e.

108

.

Case II:

When the radius of the hemispherical balloon is 12

cm,

Then,

The surface area of the balloon is

2

3 (12)

i.e.

432

.

Therefore,

The ratio of surface areas of the balloon in both cases

is

108 :432

: i.e. 1:4.

EXERCISE 13.2

Write True or False and justify your answer in each of

the following:

Question: 1

The volume of a sphere is equal to two-third of the

volume of a cylinder whose height and diameter are

equal to the diameter of the sphere.

Solution:

The height and diameter of right circular cylinder are

equal to the diameter of the sphere.

Now,

Suppose that the base radius of the right circular

cylinder is r and the height of that cylinder is 2r

whereas the radius of the sphere is r.

So,

2

11

3

2

4

3

Cylinder

Sphere

V

rh

V

r

2

4

3

r r r

rrr

3

2

Thus,

2

3

Sphere Cylinder

VV

Therefore,

The volume of a sphere is equal to two-third of the

volume of a cylinder whose height and diameter are

equal to the diameter of the sphere.

Hence,

The given statement, “The volume of a sphere is equal

to two-third of the volume of a cylinder whose height

and diameter are equal to the diameter of the sphere”

is true.

Question: 2

If the radius of a right circular cone is halved and

height is doubled, the volume will remain unchanged.

Solution:

The radius of a right circular cone is halved and the

height is doubled.

Now,

Suppose that initially base radius of the right circular

cone is 2x and height is y and later base radius of right

circular cone becomes x and height becomes 2y.

So,

2

1 1 1

2

2 2 2

V r h

V r h

22

2

x x y

x x y

2

Thus,

21

1

2

VV

Therefore,

If the radius of a right circular cone is halved and

height is doubled, the new volume is half of the

original volume.

Hence,

The given statement, “If the radius of a right circular

cone is halved and height is doubled, the volume will

remain unchanged” is false.

Question: 3

In a right circular cone, height, radius and slant height

do not always be sides of a right triangle.

Solution:

Since,

For a right circular cone,

2 2 2

r h l

Where,

l

slant height of right circular cone

r

base radius of right circular cone

h

height of right circular cone.

So,

In a right circular cone, height, radius and slant height

always be sides of a right triangle.

Hence,

The given statement, “In a right circular cone, height,

radius and slant height do not always be sides of a

right triangle” is false.

Question: 4

If the radius of a cylinder is doubled and its curved

surface area is not changed, the height must be halved.

Solution:

The radius of a cylinder is doubled and its curved

surface area is not changed.

Now,

Suppose that initially the base radius of the right

circular cylinder is x and height of that cylinder is 2y

and later the base radius of the right circular cylinder

becomes 2x whereas the curved surface area remains

unchanged.

So,

In both cases as the curved surface areas are same.

Therefore,

Curved surface area for case

I

Curved surface area

for case II

1 1 2 2

22rh r h

2

22x y xh

As a result,

2

hy

Thus,

The height of the right circular cylinder is halved.

Hence,

The given statement, “If the radius of a cylinder is

doubled and its curved surface area is not changed, the

height must be halved” is true.

Question: 5

The volume of the largest right circular cone that can

be fitted in a cube whose edge is 2r equals to the

volume of a hemisphere of radius r.

Solution:

The volume of the largest right circular cone that can

be fitted in a cube whose edge is 2r.

So,

The base radius of the right circular cone

2

2

r

r

And,

The height of right circular cone

2r

Therefore,

Volume of right circular cone

2

1

2

3

rr

3

2

3

r

Volume of the hemisphere of radius r

Hence,

The given statement, “The volume of the largest right

circular cone that can be fitted in a cube whose edge is

2r equals to the volume of a hemisphere of radius r is

true.

Question: 6

A cylinder and a right circular cone are having the

same base and same height.

The volume of the cylinder is three times the volume

of the cone.

Solution:

A cylinder and a right circular cone have the same

base and same height.

Now,

Suppose that the base radius of the right circular

cylinder is r and the height of that cylinder is h

whereas the base radius of the right circular cone is r

and the height of the cone is h.

So,

2

2

1

3

Cylinder

Cone

V

rh

V

rh

Thus,

3

Cylinder cone

VV

Therefore,

The volume of the right circular cylinder is three times

the volume of the right circular cone.

Hence,

The given statement, “If a cylinder and a right circular

cone are having the same base and same height, the

volume of the cylinder is three times the volume of the

cone” is true.

Question: 7

A cone, a hemisphere and a cylinder stand on equal

bases and have the same height. The ratio of their

volumes is 1 : 2 : 3.

Solution:

A cone, a hemisphere and a cylinder stand on equal

bases and have the same height.

Now,

Suppose that the base radius of the right circular

cylinder is r and the height of that cylinder is r

whereas the base radius of the right circular cone is r

and the height of the cone is r while the radius of the

hemisphere is r.

So,

Volume of cone: Volume of hemisphere : Volume of

cylinder

2 3 2

12

::

33

r h r r h

2 3 2

12

::

33

r r r r r

3 3 3

12

::

33

r r r

1:2:3

Therefore,

The ratio of volumes of right circular cone,

hemisphere and right circular cylinder is 1 : 2 : 3.

Hence,

The given statement, “If a cone, a hemisphere and a

cylinder stand on equal bases and have the same

height, the ratio of their volumes is 1 : 2 : 3” is true.

Question: 8

If the length of the diagonal of a cube is

63

cm, then

the length of the edge of the cube is 3 cm.

Solution:

The length of the diagonal of a cube is

63

cm.

If the side of a cube is a, the length of the diagonal of

that cube is

3.a

So,

3 6 3a

Therefore,

6a

Thus,

The length of the edge of the cube is 6 cm.

Hence,

The given statement, “If the length of the diagonal of a

cube is

63

cm, then the length of the edge of the

cube is 3 cm” is false.

Question: 9

If a sphere is inscribed in a cube, then the ratio of the

volume of the cube to the volume of the sphere will be

6:

.

Solution:

A sphere is inscribed in a cube.

Now,

Suppose that the side of a cube is 2r.

Then,

The radius of the sphere that is inscribed in it will be r.

So,

3

3

2

4

3

Cube

Sphere

r

V

V

r

3

3

2

4

3

r

r

3

3

8

4

3

r

r

6

Therefore,

The ratio of the volume of the cube to the volume of

the sphere is

6: .

Hence,

The given statement, “If a sphere is inscribed in a

cube, then the ratio of the volume of the cube to the

volume of the sphere will be

6:

” is true.

Question: 10

If the radius of a cylinder is doubled and height is

halved, the volume will be doubled.

Solution:

The radius of a right circular cylinder is doubled and

height of that cylinder is halved,

Now,

Suppose that initially the base radius of the right

circular cylinder is r and the height of that cylinder is

2y while later the base radius of right circular cylinder

becomes 2r and the height of that cylinder becomes y.

So,

2

1 1 1

2

2 2 2

V r h

V r h

2

22

r r y

r r y

1

2

Therefore,

21

2VV

Thus,

The volume of the right circular cylinder is doubled.

Hence,

The given statement, “If the radius of a cylinder is

doubled and height is halved, the volume will be

doubled” is true.

EXERCISE 13.3

Question: 1

Metal spheres, each of radius 2 cm, are packed into a

rectangular box of internal dimensions 16 cm × 8 cm ×

8 cm. When 16 spheres are packed the box is filled

with preservative liquid. Find the volume of this

liquid. Give your answer to the nearest integer.

[Use

3.14]

Solution:

Volume of the preservative liquid Volume of the

Cuboid − Volume of the 16 spheres.

Let r be the radius of the sphere. Its volume will be

3

4

3

r

Let l, b and h respectively be the length, breadth and

height of the rectangular box. Its volume will be

.l ×b×h

Volume of the preservative liquid

Volume of the Cuboid − Volume of the 16 spheres.

3

4

16

3

l b h πr

3

64

16 8 8 3.14 2

3

3.14

64 8 2

3

512

2.86

3

3

1464.32

cm

3

3

488.11cm

Question: 2

A storage tank is in the form of a cube. When it is full

of water, the volume of water is 15.625 m

3

. If the

present depth of water is 1.3 m, find the volume of

water already used from the tank.

Solution:

Let a be the side of the cube. The volume of the cube

will be a

3

.

The volume of the cube is 15. 625 m

3

.

Therefore,

3

15.625a

3

15.625a

2.5ma

The volume of the water already used from the tank

=aa

height of the tank not filled with water

2.5 2.5 1.2

3

7.5m

Question: 3

Find the amount of water displaced by a solid

spherical ball of diameter 4.2 cm, when it is

completely immersed in water.

Solution:

Let r be the radius of the sphere.

As given,

Diameter 4.2

2.1

22

r

The volume of water displaced

The volume of the sphere

3

4

3

πr

4 22

2.1 2.1 2.1

37

3

38.808cm

Question: 4

How many square metres of canvas is required for a

conical tent whose height is 3.5 m and the radius of

the base is 12 m?

Solution:

l

2

= h

2

+ r

2

l

2

= 3.5

2

+ 12

2

= 156.25

l = 12.5 m

The curved surface area of the canvas (conical shape)

of radius r and slant height

l πrl

22

12 12.5

7

3300

7

3

471.43m

Question: 5

Two solid spheres made of the same metal have

weights 5920 g and 740 g, respectively. Determine the

radius of the larger sphere, if the diameter of the

smaller one is 5 cm.

Solution:

Volume of a sphere of radius

3

4

3

r πr

Let r

1

and r

2

be the radii and V

1

and V

2

be the volumes

of the large and the small sphere respectively.

Since both the spheres are made up of the same metal,

their weights will be in proportion to their volumes.

Therefore,

1

2

5920

740

V

V

3

1

3

2

4

296

3

4

37

3

πr

πr

3

1

296 5 5 5

37 2 2 2

r

33

3

1

5r

Radius of the larger sphere = 5 cm.

Question: 6

A school provides milk to the students daily in a

cylindrical glass of diameter 7 cm. If the glass is filled

with milk up to a height of 12 cm, find how many

litres of milk is needed to serve 1600 students.

Solution:

Milk required for 1600 students

the1600 capacity of glass

1600 volumeof aglass

Volume of a cylinder of radius r and height h

2

πr h

Therefore,

Milk required for 1600 students

2

1600 πr h

22 7 7

1600 12

7 2 2

3

739200cm

739200

1000

l

739.2l

Question: 7

A cylindrical roller 2.5 m in length, 1.75 m in radius

when rolled on a road was found to cover the area of

5500 m

2

. How many revolutions did it make?

Solution:

Curved surface area of a cylinder of radius r and

height h

2πrh

Here,

1.75mr

2.5mh

The area covered by the cylinder = 5500 m

2

The area covered in 1 revolution

= The curved surface area of the cylinder

2πrh

22

2 1.75 2.5

7

2

27.5m

The number of revolution made by the cylinder

=

Area covered by thecylinder

Curved surfaceareaof thecylinder

5500

27.5

200

Question: 8

A small village, having a population of 5000, requires

75 litres of water per head per day. The village has got

an overhead tank of measurement 40 m × 25 m × 15 m.

For how many days will the water of this tank last?

Solution:

Water required per day

=Population of the village

×

Water requirement per

head per day

5000 75

375000 l

3

375000

m

1000

The volume of a rectangular tank of length l, breadth b

and height h

l b h

40 25 15

3

15000m

Required number of days

tankVolumeof the water in the

Volumeof water requirement per day

15000

375

40

Question: 9

A shopkeeper has one spherical laddoo of radius 5 cm.

With the same amount of material, how many laddoos

of radius 2.5 cm can be made?

Solution:

Volume of a sphere of radius r

3

4

3

πr

Let r

1

and r

2

be the radii of the bigger and the smaller

laddoo respectively.

Number of laddoos that can be made from the bigger

laddoo

Volumeof the bigger laddoo

Volumeof thesmaller laddoo

3

1

3

2

4

3

4

3

πr

πr

555

2.5 2.5 2.5

222

8

Question: 10

A right triangle with sides 6 cm, 8 cm and 10 cm is

revolved about the side 8 cm. Find the volume and the

curved surface of the solid so formed.

Solution:

The solid so formed is a cone.

Volume of a cone of radius r and height

2

1

3

h πr h

Curved surface area of a cone of radius r and slant

height

l πrl

Since, the right triangle with sides 6 cm, 8 cm and 10

cm is revolved about the side 8 cm, the radius of the

cone will be 6 cm and the height will be 8 cm.

The slant height of the cone will be

22

8 6 10cm

Therefore,

The volume of the right circular cone

2

1

3

πr h

1 22

668

37

2112

7

3

301.71cm

The curved surface area

22

6 10

7

1320

7

2

188.57cm

EXERCISE 13.4

Question: 1

A cylindrical tube opened at both the ends is made of

iron sheet which is 2 cm thick. If the outer diameter is

16 cm and its length is 100 cm, find how many cubic

centimeters of iron has been used in making the tube?

Solution:

Volume of iron used

22

πh R r

22

22

100 8 6

7

2200

8 6 8 6

7

2200

2 14

7

3

8800cm

Question: 2

A semi-circular sheet of metal of diameter 28 cm is

bent to form an open conical cup. Find the capacity of

the cup.

Solution:

The length of the arc

πr

22

14

7

44 cm

The circumference of base of the cone

= Length of the arc

= 44 cm

Therefore, the circumference of the cone is

2 44πr

22

2 44

7

r

7cmr

The slant height (l) of the cone = 14 cm

Height of the cone,

22

= h l r

22

= 14 7

2 2 2

= 7 2 1

=7 4 1

=7 3 cm

The volume of the conical vessel

1 22

= 7 7 7 3

37

3

1078 3

cm

3

Question: 3

A cloth having an area of 165 m

2

is shaped into the

form of a conical tent of radius 5 m.

(i) How many students can sit in the tent if a student,

on an average, occupies

2

5

m

7

on the ground?

(ii) Find the volume of the cone.

Solution:

The area of the cloth

=The curved surface area of the tent

So,

165πrl

22

5 165

7

l

21

m

2

l

The number of students who can sit inside

Areaof the base

Areaoccupied by astudent

2

5

7

πr

22 7

55

75

110

Height of the tent

22

h l r

2

2

21

5

2

441 100

4

341

2

Volume of air in the tent

2

1

3

πr h

1 22 341

55

3 7 2

3

275

341m

21

3

241.82m

Question: 4

The water for a factory is stored in a hemispherical

tank whose internal diameter is 14 m. The tank

contains 50 kilolitres of water. Water is pumped into

the tank to fill to its capacity. Calculate the volume of

water pumped into the tank.

Solution:

The capacity of the tank

3

2

3

πr

2 22

777

37

3

2156

m

3

3

2156

kl 1m 1kl

3

Water available into the tank = 50 kl

The volume of water required to be pumped in to fill

the tank

=Capacity of the tank − Water available in the tank

2156

50

3

2156 150

3

2006

3

668.67 kl

Question: 5

The volumes of the two spheres are in the ratio 64 :

27. Find the ratio of their surface areas.

Solution:

Let the volume of the first sphere be V

1

and volume of

the second sphere be V

2

.

So,

1

2

64

27

V

V

3

3

1

3

2

4

4

3

4

3

3

πr

πr

1

2

4

....... 1

3

r

r

The ratios of their surface area

2

1

2

2

4

4

πr

πr

2

1

2

r

r

2

4

3

16

9

Question: 6

A cube of side 4 cm contains a sphere touching its

sides. Find the volume of the gap in between.

Solution:

The gap can be found by subtracting the volume of the

sphere from the volume of the cube.

Volume of the cube = 4

3

Volume of the sphere

4 22

222

37

Difference in the volume

3

4 22

4 2 2 2

37

3

11

41

21

21 11

64

21

64 10

21

640

21

3

30.48cm

Question: 7

A sphere and a right circular cylinder of the same

radius have equal volumes. By what percentage does

the diameter of the cylinder exceed its height?

Solution:

Let the radius, diameter and the height of the right

circular cylinder be r, d and h respectively.

Since the volume and the radius of the sphere and the

right circular cylinder are same, therefore

32

4

3

r r h

4

3

hr

2

2

3

hr

2

3

hd

Difference between the height and the diameter

2

33

d

d h d d

Percentage by which the diameter of the cylinder

exceed its height

Difference between the height and the diameter

Height

100

3

100

2

3

50

d

d

Question: 8

30 circular plates, each of radius 14 cm and thickness

3 cm are placed one above the another to form a

cylindrical solid. Find:

(i) The total surface area

(ii) Volume of the cylinder so formed.

Solution:

Let r be the radius and h be the height of the 30

combined plates.

So,

cm14 , 3 30 90r cm h

The total surface area of the combined 30 circular

plates

2 ( )r r h

22

2 14 (14 90)

7

88 104

cm

2

9152

The volume of the cylinder so formed

2

rh

22

14 14 90

7

cm

3

55440