# Volume Of Solids (Cube, Cuboid, Prism, etc.) With Example

In this post, we will be learning about volume of solids questions and answers. We will see many volume of solids questions and answers with tricks/shortcuts etc.

So, let’s start the learning now.

## Getting Started

This post contains three sections –

• Section A – This section contains important formulas, short tricks, concepts on volume of solids.
• Section B – This section contains some examples that demonstrates how to solve questions on volume of solids using short tricks, formulas shown in section A.
• Section C – Go to this section to solve some problems on volume of solids.

Now,

Where to start from?

We recommend go through section A, then section B, then section C. Even if you are familiar with this topic, do not just skip section A and section B.

## Section A: (Terms & Formulas)

Volume: The Volume of a solid is the number of unit cubes it takes to fill up a solid. There are different types of solids for which volume can be found out.

These are mentioned below:

Shape Volume
Cube Volume of Cube = s3
Where,
s = Length of the side

###### Example:

Find the volume of cube whose side is 3 cm.

Solution:
Consider below diagram where each side is of 3 cm. So,
Volume of cube (v) = s3
Where, s = 3 cm.
v = (3)3
= 27 cm3

Cuboid Volume of cuboid = l * w * h

Where,
l = Length of cuboid,
w = Width of cuboid,
h = Height of cuboid

Example:
Find the volume of cuboid having dimensions 3 cm, 4 cm and 7 cm.

Solution: Volume of cuboid = 3 * 4 * 7 = 84 cm³

Prism  Volume of Prism = (Area of base) * height

Where,
Height = Height of prism,

Area of base depends on type of prism.

For rectangular prism,
Area of base = Length * Width

For Triangular prism,
Area of base = Area of triangular base = (1/2) * height of triangle * Length of base.

###### Example:

Find the volume of triangular prism given below – Solution:
Note that sides of triangle( having sides 3 cm and 6 cm) are perpendicular to each other.
So, area of triangle = (1 / 2) * base * height.
So, Area of the triangle = 1/2 * 3 cm * 6 cm = 9 cm²

Volume(v) of the prism = (Area of base) * height
Where,
Area of base = area of triangular base.
height = Height of prism.
So,
=> v =  9 cm * 8 cm = 72 cm³

Cylinder Volume of cylinder = π * r² * h

Where,
r = Radius of cylinder,
h = Height of cylinder

###### Example:

Find the volume of cylinder of radius 3 cm and height 5 cm.

Solution:

Volume of cylinder = π * r² * h,
Here, r = 3 cm
h = 5 cm.

So, V = π * 3² * 5,
= 3.14 * 45
= 141.37 cm3

Hollow Cylinder Volume of Hollow cylinder
= π * h * (R² – r²)
Where,
R = Radius of the outer cylindrical surface,
r = Radius of the inner cylindrical surface,
h = Height of the hollow cylinder

###### Example:

Find the amount(in cm3) of metal used in making a hollow cylinder having thickness 2 cm, outer radius 14 cm and height 70 cm.

Solution: The thickness of the hollow metallic cylinder  = 2 cm.
Height of the cylinder = 70 cm
Outer radius of the cylinder = 14 cm.
Inner radius of the cylinder = 14 – 2 = 12 cm
Volume of the metal used in making the cylinder = Volume of the hollow cylinder
= π * (R2 – r2) * h
= π * (142 – 122) * 70
= π * 52 * 70
= 3.14 * 52 * 70
= 11429.6 cm3

Cone Volume of cone = (1 / 3) * π * r² * h
Where,
r = Radius of circular base,
h = Height of cone

###### Example:

Find the volume of cone having radius 3 cm and height 11 cm.

Solution:
Below image clearly depicts the scenario – Volume (v) of cone = (1 / 3) * π * r² * h
Where,
r = 3 cm,
h = 11 cmSo, v = (1 / 3) * 3.14 * 3² * 11
= (1 / 3) * 3.14 * 99
= 103.69 cm3

Pyramid Volume of pyramid = (1 / 3) * B * h

Where,
B = Area of base,
h = height of pyramid

###### Example:

Find the volume of pyramid given below – Solution:
Volume of pyramid = (1 / 3) * B * h

Where,
B = Area of base,
h = height of pyramid
Area of base = Length * Width

Given,
Length of pyramid = 8 km,
Width of pyramid = 7 km,
Height of pyramid = 7 km
So,
Area of base = 8 * 7 = 56 km2
Volume of pyramid = (1 / 3) * (Area of base * height)
= (1 / 3) * 56 * 7 = 130.67 km3

Sphere Volume of Sphere = (4 / 3) * π * r³
Where,
r = Radius of sphere

###### Example:

Find the volume of sphere having radius 7 inch.

Solution: Volume of sphere = (4 / 3) * π * r³
So,
Volume = (4 / 3) * 3.14 * 7³
= 1436 in³

Hemisphere Volume of Hemisphere = (2 / 3) * π * r³
Where,
r = Radius of Hemisphere

###### Example:

Find the volume of hemisphere having radius 15 inch.

Solution:
A hemisphere of radius 15 inch would look like below – As we know,
Volume(v) of hemisphere = (2 / 3) * π * r³
Where, r = 15 inch

So,
v = (2 / 3) * π * 15³
= (2 / 3) * (22 / 7) * 15³
= 7,071.42 inch3

## Section B: (Examples)

Now, we will see how to approach any questions on volume of solids. What are the key factors we should look into before trying to solve such questions. Please do not move to next section without going through each volume of solids questions and answers.

#### Example 1

A rectangular prism has a volume of 4200 mm3 . It has a rectangular base of length 10 mm and width 6 mm. Find the height of the prism.

Solution

Below image clearly depicts the scenario, Given,
Volume of prism = 4200 mm3 ,
Length of prism = 10 mm,
Width of prism = 6 mm

Volume of prism = (Area of Base * height)
Area of Base = Length * Width

So,
=> 4200 = (l * w) * h
=> 4200 = (10 * 6) * h
=> 4200 = 60 * h
h = 70 mm

#### Example 2

The length of a side of a cube is equal to 8 cm. Find the Volume of the cube.

Solution

Volume(v) of cube = s3
Where, s is side of a cube

Side(s) of cube = 8 cm
So, volume = 83
= 8 * 8 * 8

Volume = 512 cm3

#### Example 3

Find the thickness of the hollow cylinder of height 100 cm if the volume between the inner and outer cylinders is 13000π mm3 and outer diameter is 14 mm.

Solution

At first, let’s convert dimensions into common unit. So, we will convert it into mm unit (10 mm. = 1 cm.)
Height of cylinder = 100 cm = 1000 mm
Outer diameter (d) = 14 mm
So, Outer radius (R) = 7 mm (Diameter = 2 x Radius)
Below diagram depicts the scenario.

Here,

r = Inner radius of cylinder,
R = Outer radius of cylinder,
h = Height of cylinder

So, we have to find R – r . Volume (V) of Cylinder = 13000π mm3
V = π * h * ( R2 – r2 )
13000π = π * 1000 * (72 – r2 )
72 – r2 = 13
49 – r2 = 13
r2 = 49 – 13
r2 = 36
r = 6 mm

So, Thickness of Cylinder = R – r
= 7 – 6

= 1 mm

#### Example 4

Find the volume of cone with radius 5 cm and height 10 cm.

Solution
Volume  (V) of cone = (1 / 3) * π * r2 * h
Where,
r = radius of base,
h = height of code

Given,
r = 5 cm,
h = 10 cm,
π = 22 / 7 So, Volume = (1 / 3) * (22 / 7) * ( 52 ) * 10

Volume of cone = 261.90 cm3

## Section C: (Exercises)

In this section, we will see different volume of solids questions and answers (with short tricks, if available).

Mark against the correct answer:

At first, please try to solve question by yourself, then move to see it’s answer.

#### Question 1

The radius of a sphere is 3r, then it’s volume will be –
a. (4 / 3)πr3
b. 36πr3
c. (8 / 3)πr3
d. (32 / 3)πr3

#### Question 2

The total surface area of a cube is 96 cm2. The volume of cube is –

a. 16 cm3
b. 512 cm3
c. 64 cm3
d. 27 cm3

#### Question 3

A cube of side 4 cm contains a sphere touching its sides. Find the volume of gap in between.

a. 30.48 cm3
b. 54 cm3
c. 46 cm3
d. 29 cm3

#### Question 4

Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is completely immersed in water.

a. 25.6 cm3
b. 38.80 cm3
c. 43 cm3
d. 56.60 cm3

#### Question 5

A shopkeeper has one jamun of radius 5 cm. With the same amount of material, how many jamun of radius 2.5 cm can be made?

a. 5
b. 10
c. 8
d. 6

#### Question 6

A right triangle with sides 6 cm, 8 cm, and 10 cm is revolved about the side 8 cm. Find the volume of the solid.
a. 330.6 cm3
b. 185.9 cm3
c. 200 cm3
d. 301.7 cm3

#### Question 7

A cylindrical tube which is opened at both the ends is made of iron sheet which is 2 cm thick. If the outer diameter is 16 cm and length is 100 cm, find how many cubic centi meters of iron has been used in making tube ?

a. 8800 cm³
b. 9000 cm³
c. 5000 cm³
d. 3000 cm³

#### Question 8

A steel vessel has a base of length 30 cm and breadth 15 cm. Water is poured in the vessel. A cubical steel box having edge of 15 cm is immersed in the vessel. How much will the water rise?

a. 7.5 cm
b. 10 cm
c. 15 cm
d. 30 cm

#### Question 9

A cylinder is having radius 1 m and height 5 m is completely filled with juice. In how many conical flasks can this juice be filled if the flask radius and height is 50 cm each?

a. 50
b. 600
c. 120
d. 90

#### Question 10

2 cm thick 15 metal plates are kept exactly one above the other. On top of the metal plate, a hemisphere of diameter 6 cm is kept. This just covers the top plate. What is the volume of the entire object?

a. 360π cm3
b. 144π cm3
c. 81 cm3
d. 288π cm3

#### Question 11

A metallic sheet is of rectangular shape with dimensions 48 m x 36 m. From each corner of the sheet, a square is cut off so as to make an open box. If the length of the square is 10 m, the volume of the box (in m3) is:
a. 4480
b. 5120
c. 6420
d. 8960

#### Question 12

The surface of a cube is 24 cm2 . Find its volume ?
a. 8 cm3
b. 16 cm3
c. 32 cm3
d. 40 cm3

#### Question 13

The areas of the two spheres are in the ratio 1 : 16. The ratio of their volume is ?
a. 1 : 4
b. 1 : 16
c. 1 : 8
d. 1 : 64

#### Question 14

A cuboidal garage has a volume of 480 m³ a length of 10 m and a width of 8 m. What is the height of the garage?
a. 4 m
b. 5 m
c. 6 m
d. 7 m

#### Question 15

Which set of dimensions belongs to a right rectangular prism with a volume of 550 cm3 ?

a. 6 cm , 12 cm , 5 cm
b. 10 cm , 5 cm , 11 cm
c. 7  cm , 9 cm , 6 cm
d. 12 cm , 2 cm , 5 cm

#### Question 16

A standard dice is a cube having sides of 10 mm. What is the volume of the dice?
a. 30 mm³
b. 45 mm³
c. 425 mm³
d. 1000 mm³

#### Question 17

A cylindrical bowl has a radius of 4.3 cm and a height of 11.6 cm. What is the volume of the bowl to the nearest tenth of a cubic centimetre?

a. 49.9 cm³
b. 168.4 cm³
c. 673.5 cm³
d. 1816.8 cm³

#### Question 18

A metallic rod has a diameter of 17 cm and has 72 cm between the two ends. What is the volume of the space between the ends?
a. 10300.4 cm
b. 3589.5 cm3
c. 67890.4 cm3
d. 16334.2 cm3

#### Question 19

A hockey puck has a diameter of 7.6 cm and a height of 3.4 cm. What is the volume of a cylindrical package containing 5 pucks?

a. 770.8 cm³
b. 480.7 cm³
c. 570.3 cm³
d. 690.4 cm³

#### Question 20

A swimming pool 9 m wide and 10 m long is 1 m deep on the shallow side and 5 m deep on the deeper side. Its volume is:

a. 390 m³
b. 270 m³
c. 420 m³
d. 530 m³

#### Question 21

A circular well with a diameter of 2 meters, is dug to a depth of 21 meters. What is the volume of the earth dug out?

a. 46 m3
b. 79 m3
c. 66 m3
d. 28 m3

#### Question 22

Two right circular cylinders of equal volumes have their heights in the ratio 1 : 4. Find the ratio of their radii.

a. 2 : 1
b. 4 : 1
c. 1 : 3
d. 1 : 2

#### Question 23

The radius of the base of a cone is 4 cm and its volume is 68.32 cm³. (π = 3.14) What is the height of the cone?

a. 16.34 cm
b. 28.48 cm
c. 47.39 cm
d. 12.48 cm

## Conclusion

We have seen how to find volume of solids like cube, cuboid, prism, cylinder, hollow cylinder, cone, sphere and hemisphere.

List of important formulas for volume of solids –

 Shape Volume Cube Volume of Cube = s3 Where, s = Length of the side Cuboid Volume of cuboid = l * w * h Where, l = Length of cuboid, w = Width of cuboid, h = Height of cuboid Prism Volume of Prism = (Area of base) * height Where, Height = Height of prism, Area of base depends on type of prism. For rectangular prism, Area of base = Length * Width For Triangular prism, Area of base = Area of triangular base = (1/2) * height of triangle * Length of base. Cylinder Volume of cylinder = π * r² * h Where, r = Radius of cylinder, h = Height of cylinder Hollow Cylinder Volume of Hollow cylinder = π * h * (R² – r²) Where, R = Radius of the outer cylindrical surface, r = Radius of the inner cylindrical surface, h = Height of the hollow cylinder Cone Volume of cone = (1 / 3) * π * r² * h Where, r = Radius of circular base, h = Height of cone Pyramid Volume of pyramid = (1 / 3) * B * h Where, B = Area of base, h = height of pyramid Sphere Volume of Sphere = (4 / 3) * π * r³ Where, r = Radius of sphere Hemisphere Volume of Hemisphere = (2 / 3) * π * r³ Where, r = Radius of Hemisphere