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Volume of a Cube

In this section, we will learn the formula of volume of a cube and how to find the volume of a cube.

A cube is a three-dimensional solid shape whose length, breadth and height are equal. It has six square faces. Each face of a cube has a side of equal length. Dice is the best example of a cube. The following figure shows the shape of a cube.

Volume of a Cube
  • Edge: A-line segment contend with two vertices is called edge. There are a total of twelve edges in a cube. These edges are of equal length.
  • Face: Faces are the square sides of a cube. There are a total of six faces (top, bottom, right, left, front, and back) in a cube.
  • Vertex: A point where three edges meet is called the vertex. There are a total of eight vertices in a cube.

The volume of a Cube

The number of cubic units that a cube occupied is called the volume of the cube. It is the product of length, breadth, and height. In other words, it is the cube of one side. It is denoted by the letter V.

The Formula of Volume of a cube

Multiply the length (l), breadth (b), and height (h) together to get the volume of a cube. Remember that length, breadth, and height must be equal in size.

The Volume of a Cube (V)=length×breadth×height

Or

The Volume of a Cube (V)=l×b×h

Suppose the length, breadth, and height of a cube is a, the volume will be:

Volume of a Cube
The Volume of a Cube (V)=a×a×a

Or

The Volume of a Cube (V)=a3

Where:

V: is the volume

a: is a side of the cube

When the length of the diagonal is given

Volume of a Cube

Suppose, the diagonal length is d, then the volume of the cube will be:

Volume of a Cube

Where:

V: is the volume

d: is the length of diagonal

Derivation of the Formula

The space occupied by a solid object is called the volume of that object. We know that all the sides (edges) in a cube are of equal length. Therefore, the formula of the volume of a cube can be derived as follows:

  • Take a cardboard of square shape.
  • Find the area of that cardboard by multiplying length and breadth together.
  • As we have taken a square piece of the cardboard, it means the length and breadth will be equal. Suppose, the length and breadth are a, then the surface area of the cardboard will be a2.
  • To get a cubical shape, we will stack multiple cardboard on that piece, one on each other. Now, we can find the height of the cube.
  • To get the volume of the cube, multiply the surface area of the cardboard by the height.
  • Form the above steps, we can conclude that the area covered by the cube is the product of the surface area of a square and height.

Let's see how to find the volume of the cube.

Example 1: A side of a cube is 9 cm. Find the volume of a cube.

Solution:

Given, side = 9 cm

volume (V)=?

Volume of a Cube

According to the formula:

The volume of a cube (V) = a3

Putting the value of side in the above formula, we get:

V= 93
V= 729

Hence, the volume of the cube is 729 cm3.

Example 2: The diagonal length of a gift box is 7 cm. Find the volume of the box.

Solution:

Given, diagonal length (d) = 7 cm

volume (V) =?

Volume of a Cube

According to the formula:

Volume of a Cube

Putting the value of d in the above formula, we get:

Volume of a Cube

Hence, the volume of the cube is 66 cm3.

Example 3: The volume of dice is 64 cm3. Find the length of the edge of the dice.

Solution:

Given, volume (V) = 64 cm3

side (a)=?

According to the formula:

The volume of a cube (V) = a3

Putting the value of side in the above formula, we get:

64= a3
64= a
a=4

Hence, the length of an edge of the dice is 4 cm.

Example 4: Find the volume of the cube given below.

Volume of a Cube

Solution:

Given, side (a) = 4.5 cm

volume (V)=?

According to the formula:

The volume of a cube (V) = a3

Putting the value of side in the above formula, we get:

V = (4.5)3
V = 91.125≈91.13

Hence, the volume of the given cube is 91.13 cm3.






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