# Surface Area of a Cube

Sub Topics
The prisms are formed with two parallel congruent polygonal bases. The lateral faces of a prism are parallelograms. In the case of a right prism, the lateral surfaces are all rectangles.

A rectangular prism or a cuboid has rectangles as bases. It has six rectangular faces, consisting of two base rectangles and four lateral rectangles. It has 12 edges and 8 vertices.
A cube is a special type of rectangular prism with all six square faces. Just as a rectangular prism, a cube also has 12 edges and 8 vertices.

Net for a cube: If a cubic box is cut suitably at the edges and laid flat, a pattern for a cube can be seen. This is called the net for the cube. The net pattern can be folded along the edges as shown and made into a cube without any overlap.

Surface area of a cube using net:
The surface area of a prism  = 2 base area + lateral surface area.
The net of the cube shows that the surface of the cube contains six congruent squares.  Any of the two opposite squares can be termed as the bases, while the remaining vertical square faces form the lateral surface. Hence the surface area of a cube is six times the area of its base. If the length of an edge is a units then the area of the base square = $a^{2}$ sq units.

The surface area of a cube - Formula
Surface area of cube of edge length a units = 6$a^{2}$ sq.units.

Areas are expressed as square units and volume as cubic units.Inversely if the surface area 'A' of a cube is known, the length of the edge is given by,
Length of the edge of a cube =$\sqrt{\frac{s}{6}}$ units.

## How to find the Surface Area of a Cube

The surface area of a cube is the combined area of all six of the sides on its surface. All six sides of a cube are congruent, so to find the surface area of a cube,find the surface area of one side of the cube and then multiply it by six.

Formula
Surface area of cube = 6$a^{2}$ sq.units.
Where a= length of one side

## Surface Area and Volume of a Cube

The net of a solid shows the surface spread out, while the solid build out of the net occupies space. While the surface area refers to the area of the material required to form the solid, volume is the measure of space that the solid occupies. The formula for the volume of a cube is,
Volume of a cube – Formula
Volume of a cube of edge length a units = a3 cubic units.

The surface area can be determined from the given volume and inversely volume can be found from surface area after determining the edge of the cube.

## Surface Area of a Cube Problems

### Solved Examples

Question 1: Find the surface area of the cube with the volume of 512 $cm^{3}$ ?
Solution:

Given : volume =512 $cm^{3}$

Volume of cube = $(side \ length)^{3}$
=>512 $cm^{3}$ = $(side)^{3}$
=>side = cube root of 512

since 512 $cm^{3}$ = 8 cm $\times$ 8 cm $\times$ 8 cm

cube root of 512 = 8
Hence the side length of the cube is 8 cm.

Now,
surface area of cube = 6 $(side)^{2}$ = 6 $(8 \ cm)^{2}$ = 6 X 64 $cm^{2}$ = 384 $cm^{2}$

Question 2:
Each edge of a cube is increased by 10%. Find the percentage increase in the surface area of the cube.

Solution:

Let side of cube = x
surface area = 6 $x^{2}$

new side = x $\times$ $\frac{110}{100}$ = $(1.1 \ x)^{2}$
new surface area = 6 $\times$ $(1.1 \ x)^{2}$ = 6 $(1.21 \ x)^{2}$ = 7.26

increase in area = 7.26 $x^{2}$ – 6$x^{2}$ = 1.26 $x^{2}$
% increase = $\frac{1.26 \ x^{2}}{6x^{2}}$ $\times$ 100

the percentage increase in the surface area of the cube= 21%