# Lateral Surface Area of Cube

## Definition of Lateral Surface Area of a Cube

The term “Lateral Surface” refers to the side faces of any three-dimensional object, excluding its bottom and the top. Therefore, the Lateral Surface Area (LSA) of a Cube may be defined as the total area of the four side faces excluding the bottom and the top face.

## Explanation of Lateral Surface Area of a Cube

The cube is a three-dimensional shape with 6 faces, 4 at the sides, 1 on the top and 1 on the bottom, as shown in the figure below. Since the dimensions of the four sides are the same, it can be said that a cube is composed of six squares. To understand the concept of the Lateral Surface Area (LSA) of a Cube let us consider a Cube as shown in the figure below.

As shown in the figure above, the cube has six faces, called squares, and the length of the edges is indicated by the letter $a$.

Now, if we open the above cube, then the six faces of the cube will look like the following figure,

The Area of each face of the cube is, $A=a^2$

Since, the Lateral Surface Area only considers four side faces, the Lateral Surface Area of the cube will be,

$LSA=a^2+a^2+a^2+a^2$

$\therefore\,\,\,\, LSA=4a^2$

Which is the formula for finding the Lateral Surface Area (LSA) of a Cube.

## How to Find the Lateral Surface Area of a Cube?

As discussed earlier in the above section, the Lateral Surface Area (LSA) of a Cube is the sum of the areas of the four side faces. The same can be obtained by using the following two methods.

### When the Edges of a Cube are Given

When the length of the edges of a cube is given then the Formula for finding the LSA of the Cube is given by,

$LSA=4a^2$

Where, $a$ is the length of the edges of the cube.

### When the Total Surface Area of a Cube is Given

The Total Surface Area (TSA) of a Cube is the total area of all the six sides and is given by,

$TSA= 6a^2$

$\therefore\,\,\,\, a^2=\frac{TSA}{6}$

Therefore, the LSA is,

$LSA=4a^2$

$LSA=4\times \frac{TSA}{6}$

$LSA = \frac {2}{3}\times TSA$

## How to Calculate Lateral Surface Area of a Cube with Diagonal

If the length of the edges of a cube are not given but the length of the diagonal$(d)$ is given, then the Lateral Surface Area (LSA) of the Cube can be calculated by using the following formula,

$LSA=\frac{4d^2}{3}$

## Lateral Surface Area of Cube Formula

All the above formulas for finding the lateral surface of area of a cube are tabulated below:

## Solved Examples

1. A cubical box has each edge of 20 cm. Find the lateral Surface Area of the Box?

Solution: Given, $a=20\,cm$
The formula for finding the LSA of a Cube is,
$LSA=4a^2$
$LSA=4\times 20^2$
$LSA=4\times 400$
$LSA=1600\,cm^2$
Therefore, the Lateral Surface Area of the cubical box is $1600\,cm^2$

2. Find the lateral surface area of the cube with area of one face $81cm^2$ also find the length of the side?

Solution:
Method 1:
Given the area of one face=$81\,cm^2$
The area of one face of a cube is given by,
$A=a^2$
Therefore, $a^2=81\,cm^2$
Length of the side, $a=9\,cm$
Now, the LSA of the cube will be,
$LSA=4a^2$
$LSA=4\times 9^2$
$LSA=4\times 81$
$LSA=324 cm^2$

Method 2:
Given the area of one face, $a^2=81\,cm^2$
Length of the side, $a=9\,cm$
The LSA of a cube is the area of four side faces, therefore,
$LSA=4\times a^2$
$LSA=4\times 81$
$LSA=324\,cm^2$