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Volume of a Cube

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Volume of a cube is defined as the total number of cubic units occupied by the cube completely. A cube is a three-dimensional solid figure, having 6 square faces. Volume is nothing but the total space occupied by an object. An object with a larger volume would occupy more space. The volume of the cube is calculated by multiplying the length, breadth, and height of the cube. For a cube the length, breadth, and height are equal. Thus, the volume of a cube is just side cube.

Let us understand the volume of a cube in detail along with the formula and solved examples in the following sections. Also, learn about the Surface area of the cube here.

What is Volume of a Cube?

Volume of a cube is defined as the total capacity of the cube it is the total amount of liquid a cube can hold. The volume of a cube is measured in cubic units such as cm3, m3, etc.

A cube is a solid 3-D figure, with 6 square faces. All the faces of a cube are square hence it has all dimensions equal

Let the length, width, and height of a cube be ‘a’, then;

Volume of cube = a × a × a

Volume of Cube = a3

All corners of a cube meet at an angle of 90° degrees. The figure below shows a cube, where l is the length, b is the width, h is the height and l = b = h. The length, width, and height represent the edges of the cube and when three edges meet at a point, it is called a vertex.

Cube Structure

 

Volume of Cube Formula

Volume of a cube is defined as the total number of cube units that the cube occupies completely. A cube is a three-dimensional shape with six faces, twelve edges, and eight vertices. Therefore, the volume of a cube is the space surrounded by its six faces. Volume of the cube is calculated using two formulas which are discussed below:

Volume of Cube If Side is Given

The formula to calculate the volume of a cube when the side (Let a) of the cube is given

Volume of the cube = a × a × a 
                     = a3

Thus, when the edge length is known volume of the cube can easily be found

Example: Find the volume of a cube with a side of 5 cm

Solution:

Given, 

Edge length( a) = 5 cm

Volume = 53 
             = 5 x 5 x 5 
             = 125 cm3

Volume of a Cube If Diagonal is Given

The formula to calculate the volume of a cube when the diagonal of the cube is given

Volume of the cube = [√3 × (d)3] / 9

where 
d is the diagonal of cube

Volume of a Cube Equation

The equation which gives the volume of a cube is discussed below. Suppose a cube of edge length ‘a’ is taken then its volume is calculated using the formula.

Volume of Cube(V) = a × a × a = a3

For example, what is the volume of the cube if the side length is 7 m?

Solution:

Side of cube =7 m

Volume of cube equation,

v = a3

putting the value of a in above equation we get,

v = (7)3

v = 643 m3

Thus, the volume of the cone is 643 m3

Derivation of Volume of a Cube

The volume of any object is the space occupied by that solid in the 3-D plane. In a cube all the sides i.e. length, breadth, and height are equal (l = b = h). The formula for the volume of a cube is derived as follows:

  • A cube can be considered as layers of squares that are stacked on top of one another. Thus, for the base of a square shape, the area is length multiplied by its breadth.
  • In a square, length, and breadth are equal, thus the area will be “a2“.
  • A cube is made by adding multiple layers of square sheets on top of one other until the height becomes “a” unit. Thus, the height of the cube is “a”.

Now the volume of any regular figure is the base area multiplied by the height.
Thus,

Volume of Cube = Base Area × Height

                        = a2 × a = a3 units3

How to Find the Volume of a Cube?

Two methods by which the volume of a cube can be found are

  • Using Edge-length
  • Using Diagonal

Volume of a Cube is calculated using the steps discussed below:

Step 1: Note the dimension of the cube. Let the side is represented by (a)  and the diagonal is represented as (d).

Step 2: Now use the formula,

V = a3 

where 
a is the length of the side of a cube, 

OR 

V = [√3 × (d)3] / 9

where 
d is the diagonal of the cube accordingly

Step 3: Simplify the above eqaution.

Step 4: Add unit3 to the answer in step 3 to the volume of the cube.

As the volume of a cube is a cubic function it increases drastically if we change the dimension of the cube. This can be understood by the following image.

Increasing Volume of Cube

 

Surface Area of a Cube

Surface area of the cube is the total area covered by all the faces of the cube. As a cube has six square faces of similar dimensions its volume is calculated by the formula,

Surface Area of Cube = 6a2

where,
a is the side of the cube.

Learn More, Surface Area of a Cube

Volume of a Cube and Cuboid

Cube is a three-dimensional figure with six faces and three dimensions length, breadth, and height but for a cube all the dimension length = breadth = height = a(say). Then its volume is given as,

Volume = a3

Cuboid is a three-dimensional figure with six faces and three dimensions length, breadth, and height> Leth the length, breadth, and height of the cube are l, b, and h respectively then its volume is given as,

Volume = l × b × h

Learn more, Volume of a Cuboid

Examples of Volume of a Cube from Everyday Life

Various examples which we come across in our daily life resembles cube and we are required to find their volume. Some of the common examples are,

  • A cubical cardboard box is used to pack various objects.
  • Some of the rooms we live in are shaped like cubes.
  • An aquarium in the shape of a cube can hold water and the amount of water it can hold is calculated using the volume of cube formula, etc.

Read More

Solved Examples on Volume of Cube

Example 1: If the volume of a cube is 216 cm3, what is the dimension of the cube?

Solution: 

Given, 

The volume of a cube, V = 216 cm3

Volume of cube = (side)3 

                         = (216)= (6)3

Therefore, the side of cube is 6 cm

Example 2: How many 3 cm × 3 cm × 3 cm cube boxes can fit in a large 15 cm cube box? 

Solution: 

Volume of each box = (3 × 3 × 3) cm3 = 27 cm3.

Volume of large cube box = (15 × 15 × 15) cm3 = 3375 cm3.

Number of boxes = Volume of large cube / Volume of small cube 

                            = 3375cm3 / 27cm3 

                            = 125 boxes

Thus, 125 boxes are required to fit in the large box.

Example 3: The volume of a cubic hard disk is 0.5 dm3. What are the dimensions of the disk?

Solution: 

Since, the Volume of a cube = a3  

0.5 = a3

a = 3√0.5

   = 0.794 dm

Example 4: Calculate the volume of a cube with a diagonal of 3 inches.

Solution:

Given,

Diagonal = 9 inch.

Cube Volume = [√3 × (Diagonal)3] / 9

Volume = √3×[(3)3/9] 

              = √3 × 3 
              = 1.732 × 3

              = 5.196 inches3

Example 5: Find the edge of a cube whose volume is 1000 cm3

Solution:

Volume = 1000 cm3l

Volume = a3 

where, 
a is edge of the cube

1000 = 103 = a3

a (edge) = 10 cm

Thus, the edge of the cube is 10 cm.

Example 6: Find the volume of a cube of side 0.01 cm

Solution:

Given, 

Edge (a) = 0.01 cm

Volume = a3

Volume = (0.01)3 
             = 0.000001 cm3

Thus, the volume of the cube is 10-6 cm3

FAQs on Volume of a Cube

Q1: What is the volume of a cube?

Answer:

Volume of a cube is defined as the total capacity of a cube. It is the total amount of liquid a cube can hold.

Q2: What is the unit of volume?

Answer:

The unit of volume is given by cubic units i.e. volume is always measured in m3, cm3, etc. It is generally measured in liters.

Q3: How to find the volume of a cube?

Answer:

The volume of the cube is calculated using the formula,

Volume = a3

where
a is the edge length of a cube

Q4: What is the volume of a cube if diagonals of the cube are given?

Answer:

The volume of a cube when diagonals are given is calculated using the formula,

Volume = [√3 × (d)3] / 9

where
d is the diagonal of a cube

Q5: How many litres are in a meter cube?

Answer:

We know that,

1 cm = 1 cc (cubic centimeter) = 1 ml and,

1 m3 = 1000000 cm3

1 m3 = 1000000 ml

1 m3 = 1000 litres

Thus, 1 m3 has 1000 litres

Q6: What is the volume of a cube of side 2a?

Answer:

The formula for the volume of a cube is

Volume = (side)3

Now if the side is 2a then its volume is,

Volume = (2a)3 

              = 8a3

Thus, the volume of the cube with side 2a is 8a3



Last Updated : 18 Feb, 2024
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