Volume and Surface Area of Cube

What is Cube?

A cuboid is a cube if its length, breadth and height are equal.

In a cube, all the faces are squares which are equal in area and all the edges are equal. A dice is an example of a cube.

Volume of a Cube (V) = (edge)3 = a3

Total surface Area of a Cube (S) = 6(edge)2 = 6a2

Diagonal a Cube (d) = √3(edge) = √3a

Where a = edge

Problems on Volume and Surface Area of Cube:

1. If the edge of a cube measures 5 cm, find (i) it volume, (ii) its surface area, and (iii) the length of a diagonal.

Solution:

(i) volume = (edge)3

= 53 cm3

= 125 cm3

(ii) Surface area = 6(edge)2

= 6 × 52 cm2

= 6 × 25 cm2

= 150 cm2

(iii) The length of a diagonal = √3(edge)

= √3 × 5 cm.

= 5√3 cm.

2. If the surface area of a cube is 96 cm2, find its volume.

Solution:

Let the edge of the cube be x.

Then, its surface area = 6x2

Therefore, 96 cm2 = 6x2

⟹ x2 = $$\frac{96 cm^{2}}{6}$$

⟹ x2 = 16 cm2

⟹ x = 4 cm.

Therefore, edge = 4 cm.

Therefore, the volume = (edge)3

= 43 cm3

= 64 cm3.

3. A cube of edge 2 cm is divided into cubes of edge 1 cm. How many cubes will be made? Find the total surface area of the smaller cubes.

Solution:

Volume of the bigger cube = (edge)3

= 23 cm3

= 8 cm3.

Volume of each of the smaller cubes = (edge)3

= 13 cm3

= 1 cm3

Therefore, the number of smaller cubes = $$\frac{8 cm^{3}}{1 cm^{3}}$$

= 8

The total surface area of a smaller cube = 6(edge)2

= 6 × 1 cm2

= 6 cm2

Therefore, the total surface area of the eight smaller cubes = 8 × 6 cm= 48 cm2.