# Total Surface Area of Cube

## What is the Total Surface Area of Cube?

The phrase “Total Surface” of simply “Surface” refers to the outer faces of any three-dimensional object. Therefore, the Total Surface Area (TSA) of a Cube may be defined as the sum of the Area of all six faces of the Cube.

To understand the concept of the Total Surface Area of a Cube, let us consider a Cube as shown in the figure below,

The Cube shown in the above figure has six faces (4 side faces, 1 top face and 1 bottom face). Since, it is a Cube, the length of all the edges are equal and denoted by the letter $a$. The Total Surface Area of the Cube will be the sum of the area of all six faces.

Area of one face,

$A=a^2$

Therefore, the Area of six faces,

$TSA=6\times a^2$

Which is the formula for finding TSA of cube.

## How to find the Total Surface Area of a Cube?

The Total Surface Area of the Cube can be found out from the Edge or the Lateral Surface Area of the cube. If any of the two are given, then, we can easily find the TSA of the Cube.

### Total Surface Area of Cube when the Edge is given

If the length of the edges of a cube, denoted by the letter $a$, is given, then, the formula for finding the Total Surface Area (TSA) of the cube is given by,

$TSA=6a^2$

### Total Surface Area of Cube when Lateral Surface Area is given

If the Lateral Surface Area of a Cube is given, then the Total Surface Area of the cube can be found out from the following relation,

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

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

## Solved Examples

1. Find the total surface area of a cubic box if the length of the edge is 5cm?

Solution: Given, the length of edge, $a=5\,cm$
The total surface area of the cubic box will be,
$TSA=6\times a^2$
$TSA=6\times 5^2$
$TSA=6\times 25$
$TSA=150\,cm^2$

2. Reena wants to paint a cubical box for her school project. The edge of the box is 1.1 meter and the cost of painting per square meter is 50 Rs. Find the total cost for Painting the whole box?

Solution: Given, edge, $a=1.1\,m$
The TSA of the box is,
$TSA=6\times a^2$
$TSA=6\times (1.1)^2$
$TSA=6\times 1.21$
$TSA=7.26\,m^2$
The cost of painting per square meter is $50$ Rs. Therefore, the cost of painting the whole box,
$=7.26\,m^2\times 50\,Rs.$
$=363 Rs.$