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 cm² = 48 cm².

9th Grade Math

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