 # Volume of a Cube

## What is a Cube?

Cube is a three-dimensional figure made up of six square faces having an equal length of sides all throughout. Example- Rubik’s cube, dice.

## What is Volume of a Cube?

The total space occupied by a cube is called the volume of a cube. It is the capacity of a cube. The amount of water contained in ice cubes is equal to the volume of the cube of ice. The volume of a solid cube can also be defined as the number of unit cubes required to fill up a cube.

Unit Cube: It is a cube of length, breadth and height of one unit. The unit can either be in cm or metre.

### The volume of a cube formula

The volume of any solid object is the multiplication of its length, breadth and height. Since the measurement of the length, breadth and height of a cube are same so the volume of a cube is taken as a cube of the length of its side.

The volume = length $\times$ breadth $\times$ height

Let the side of the cube be ‘a’.

The volume of a cube = side $\times$ side  $\times$ side

= a $\times$ a $\times$ a

= ${a^3}$

## Concept of Volume of a Cube

### The volume of a hollow cube :

Let us assume a hollow cube having very thin walls such that the volume occupied by its wall is negligible. Thus, we can say that the volume occupied by the cube itself is insignificant. Now, if we fill the cube with water then the amount of water the cube can hold is the volume of the cube.

### The Volume of a Solid Cube :

The volume of a solid cube is the number of cubic units required to fill the entire cube. The volume of one cubic unit is 1cm3 so the number of unit cubes making up the cube is the volume of the cube. A solid cube can also be viewed as a bunch of multiple square sheets. If the side of the square sheet is ‘a’, then the surface area of each sheet of a square is ‘a2’. Volume of the solid cube is equal to the number of square sheets.

Let’s suppose the length and breadth of the square sheet are 10cm. So, the surface area of the cube is equal to a2 that is 10x10 = 100 cm2. If there are ten sheets each of height 1cm then,

Volume = Surface area of sheet $\times$ number of sheets

$= {a^2} \times {a^{\boxed{}}}$

= 100 x 10

= 1000$c{m^3}$

The volume of a solid cube with diagonal:

Let the diagonal of the cube be ‘D’ and the side be ‘x’

Length of the diagonal = $= \sqrt[{\boxed{}}]{{3 \times }}{\text{ }}side$

= $\sqrt[{\boxed{}}]{{3 \times }}{\text{ }}x$

Side of the cube =$\frac{{Diagonal}}{{\sqrt[{\boxed{}}]{3}}}$

= ${\left( {\frac{D}{{\sqrt[{\boxed{}}]{3}}}} \right)^3}$

The volume of the cube = side x side x side

= ${x^3}$

= ${\left( {\frac{D}{{\sqrt[{\boxed{}}]{3}}}} \right)^3}$                          (on rationalising by 3)

= $\sqrt 3 \frac{{{D^3}}}{9}$

## Summary On The Volume Of A Cube Formula

 The Volume of a Cube Formula When the length of its side is given a3(side x side x side) When the length of its diagonal is given 3D39 (D stands for diagonal) When the surface area and number of square sheets given. Surface area of sheet x number of sheets When the volume of the unit cube and its number are given Unit Volume (in cm or m) x no of unit cubes

Volume of a Cube - Solved Examples

1. Find the volume of a cube with side 4cm.

Solution:

Side of the cube = 4cm

We know,

Volume of the cube = side $\times$ side $\times$ side

=  4cm $\times$ 4cm $\times$ 4cm

= 64 $c{m^3}$

Therefore, the volume of cube = 64 $c{m^3}$

1. A company makes cubical matchbox of sides 2 cm. While shipping these matchboxes are kept in a big cubical container. If each container can accommodate 8 boxes then what is the volume of the big cubical container? Also, find the side of the container.

Solution:

Each container consists of 8 boxes.

If the side of each box is 2 cm the volume of each box is,

Volume of each box = 2$\times$2$\times$2 = 8$c{m^3}$.

Volume of 8 boxes = 8 $\times$ 8 $c{m^3}$

= 64 $c{m^3}$

The volume of the container is equal to the volume of 8 boxes. Therefore, volume of the container is 64 cm3.

Side of the container = $\sqrt{{volume}}$

= $\sqrt{{64}}$

=  4 cm

Thus, the side of the container is 4cm.

1. If the Volume of a cube is  $125inche{s^3}$. Find the length of its edge.

Solution:

The volume of cube  = $125inche{s^3}$

Side of the cube =$\sqrt{{\left( {Volume} \right)}}$

=$\sqrt{{125}}$

=  5 cm.

Therefore, the side of the cube of volume $125inche{s^3}$ is 5cm.

1. If the capacity of a cubical tank to hold water is 1331000 ml then find the length of the side of the tank in cm.

Solution:

If the capacity of the tank to hold water is equal to the volume of the tank.

The Volume of the tank = 1331000ml

We know,

1cm3 = 1 ml

Therefore, the volume of a cube is also equal to 1331000 cm3

Side of the cube =$\sqrt{{\left( {Volume} \right)}}$

=$\sqrt{{1331000}}$

=  110 cm.

Thus, the length of each side of the cube is 110 cm.

1. If the length of the side of a bigger cube is 4cm longer than the length of the side of the smaller cube and the difference in their volume is 316 cm 3. Find the length of each side of the cube.

Solution:

Let the length of the side of the smaller cube be x and larger cube be y.

If the length of the larger cube is 4 times the length of the side of the smaller cube then,

y = 4 + x

The volume of the smaller cube =  x 3

The volume of the larger cube =  y 3 =  (4+ x) 3

The difference in their volume = 316 cm 3

The volume of the larger cube - The volume of the smaller cube = 316 cm 3

y 3 -  x 3 = 316

(a + 4) 3 - a 3 = 316

Applying the formula, a3 - b3 = ( a- b)(a2 + ab + b2)

( a + 4 -a)[( a + 4) 2 + a(a + 4) + a 2 ) ] = 316

4 [a 2 + 2.a.4 + 4 2 + a 2 + 4a + a 2 ]

4[ 3a 2 + 8a + 16 + 4a]

4 [3a 2 + 12a + 16] = 316

3a 2 + 12a + 16 = 79

3a 2 + 12a - 63 = 0

a 2 + 4a - 21 = 0

( a + 7)( a - 3) = 0

∴ a = -7 or a = 3

Since the length of the side of the cube can never be negative (i.e, -7) so that means the side of the smaller cube is equal to 3cm.

Length of the side of bigger cube = 4 + length of the side of the smaller cube.

y = 4 + x

y = 4 + 3

y = 7.

Therefore, the length of the side of the bigger cube and the smaller cube is 7cm and 3cm respectively.

Q1) What are the examples of a cube?

Dice, Ice, Sugar cubes, Ice cubes, Rubik’s cube, etc.

Q2) What do you mean by diagonal of a cube?