Volume of a Cube

Introduction

There are many geometrical solids around us. For example, two kids are playing board games using dice & coins. One boy is trying to solve Rubik’s cube. One thing that is common in the above examples is that these objects are in the shape of cubes. In this tutorial, we are going to discuss the topic of cubes & volume of a cube. A cube is a three-dimensional solid object having equal sides & faces. A cube has six faces, twelve edges & vertices. It is also known as the regular hexahedron or square prism. Volume of a cube is the amount of space occupied by the cube. It is one of the essential & fundamental concepts in geometry. The concept of the volume of a cube is majorly used for finding the capacity of a cubical tank.

What is a cube?

A cube is a symmetrical three-dimensional solid object bounded by six faces, facets or sides with three meeting at the point or corner known as the vertex. The cube has six faces, twelve edges & eight vertices. The dimensions of a cube are the same. It is also known as the regular hexahedron or square prism. Cube is a platonic solid (solid having congruent faces.)

The above figure represents a cube. Dice, ice cubes & Rubik’s cubes are some real-life examples of cubes. 

Some properties of the cube :

• A cube has 12 edges, 6 faces & 8 vertices. Faces of the cubes are square, therefore length, breadth & height of the cubes are equal.
• The angles between any edges & faces are \({90^0}\).
• The opposite planes or faces are parallel to each other.
• Three edges & three planes meet at each vertex of the cube.

What is volume of cube ?

The volume of a cube is a three-dimensional space occupied by the cube. For computing, the volume of the cube has two different formulas depending upon different parameters. By knowing the length of an edge of the cube we can calculate the volume of a cube. Also, the volume of a cube can be calculated by using a length of a diagonal. the volume of a cube is expressed in cubic units. Most of the time volume of cubes is expressed in SI units  \({m^3}\) in CGS units, \({cm^3}\) & litre.

Volume of a cube can be calculated by using two methods :

1) By using edge length 

2) By using diagonal

1) By using edge length

The volume of a cube having edge length ‘l’ can be calculated as 

Volume of a cube = length x breadth x height

 \(\begin{array}{l} = l \times l \times l\\ = {a^3}\end{array}\)

Derivation for a volume of a cube:

Consider a square sheet. The area of a sheet will be taken as surface area, the area of a sheet is length x breadth.

As the sheet is square. It has equal length & breadth, therefore the surface area will be \({a^2}\).

By stacking multiple sheets on top of each other square can be formed so that we will get height ‘a’ of a cube. 

Now, we can conclude as the overall area covered by the cube will be the area of the base multiplied by the height.

So, the volume of a cube \( = {a^2} \times a = {a^3}\)

So here we can conclude as the Volume of the cube \( = {\left( {side} \right)^3} \)

For example, the Volume of a cube having a side of 3 m can be calculated as  

             \( = {\left( {side} \right)^3} = {\left( 3 \right)^3} = 27{m^3}\)

2) By using diagonal 

Volume of a cube can be calculated by using a formula,

Volume of a cube \( = \frac{{\sqrt 3  \times {d^3}}}{9}\)

Here d is the diagonal of the cube 

For example, Length of a diagonal is 3 cm, then the volume of the cube will be

\[ = \frac{{\sqrt 3  \times {d^3}}}{9} = \frac{{\sqrt 3  \times {{\left( 3 \right)}^3}}}{9} = 3\sqrt 3 c{m^3}\]

Some other important formulae of a cube:

i)Total surface area of the cube =\(6{l^2}\) units

ii)Lateral surface area of the cube =\(4{l^2}\) units

Also Read: how to calculate volume of cuboid
                  how to find the volume of a sphere

Solved examples:

Q 1) Calculate the volume of a cube if the edge length of a cube is 

i) 9 cm

ii) 5.2 cm

 iii) 7.5 cm

   ii) 5.2 cm

 iii) 7.5 cm

Q 2) Calculate the edge length of a cube if the volume of the cube is

Q 3) Compute the volume of a cube which has a total surface area of 661.5 sq. cm.

Solution: Here, the total surface area = 661.5 sq. cm and the volume of a cube =?

Using the formula for the total surface area of a cube,

Q 4) Calculate the volume of the cube having a diagonal of 5 units.

Solution: Here length of a diagonal = 5 units 

Volume of a cube by a diagonal formula is given as,

Therefore, the volume of the cube is 24.05 cubic units

Word problems :

1) A cubical tank can store 1331000 ml of water. Then compute the side of the tank in cm.

Solution: Volume of a tank = 1331000 ml

2) Calculate the number of cubes when we cut a cube with an edge of 27 cm into cubes having an edge length of 3 cm?

Summary :

In this tutorial, we have learned about cubes & how to calculate the volume of cubes. A cube is a three-dimensional solid object having equal faces & edge lengths. Volume of a cube is a space occupied by a cube. Rubik’s cube, cubical tank, cubical box, and dice are some real-life examples of cubes. The volume of a cube can be calculated by two methods. The first is by using the edge length formula & by using the diagonal formula. This concept has wide application in real life. The concept of the volume of a cube is mainly used to determine the capacity of a cubical tank & find the side of a tank. This tutorial will surely help you to understand cubes & volume of cubes.

 

Frequently Asked Questions 

1. Why cube is known as a regular hexahedron?

Ans. A regular hexahedron is a 3D solid object having six congruent faces. Cube has six congruent faces, therefore cube is known as the regular hexahedron.

2. State the difference between cube & cuboid?

Ans. A cube is a three-dimensional solid object having all square faces whereas a cuboid is a three-dimensional object having all rectangle faces.

3. Can a prism be a cube?

Ans. A cube is a prism because a cube is considered one of the platonic solids.

4. State the difference between the surface area & lateral surface area of a cube?

Ans. For the calculating surface area of a cube sum of the area of all faces is taken whereas for calculating lateral surface area sum of only four surfaces is taken.

i)Total surface area of the cube = \(6{l^2}\) units

ii)Lateral surface area of the cube =  \(4{l^2}\) units

5. Explain what is net of a cube is?

Ans. The net of a cube is formed when the square faces of the cube are flattened by separating at the edges to form a 2D figure. Through that figure, we can see six faces of the cube.

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