Surface Area of Cube

What is a Cube?

A cube is a three-dimensional solid figure bounded by six square faces. It has 6 faces, 8 vertices and 12 edges. 

Example: Dice, Ice cube, Rubik's cube, etc.

In this article, you will learn about the surface area of the cube, its formula and derivation along with some solved examples.

Surface Area of Cube

In geometry, the area of any shape is the region covered by it in a plane. A cube consists of 6 square faces, the region covered by these square units on the surface of the cube is called the surface area of the cube. So, the surface area of the cube is the sum of the area of its six square faces.

Formula of Surface Area of Cube 

Let us consider a cube whose each edge or side has length of ‘a’ units. 

The area of one of the six faces of the cube is the area of a square with side ‘a’ units.

i.e. Area of one face of cube = Area of square = side × side = (side)2 = a2 sq. units

Therefore, the total surface area of cube = sum of area of its six square faces

Since the area of each of six square faces are the same, the sum of the area of its six square faces can be obtained by multiplying 6 to the area of one face of the cube.

Hence, the total surface area of cube = 6 × (a2) sq. units

Or, Total surface area (TSA) of cube = 6a2 sq. units

Length of Edge of Cube

The length of edge of the cube can be obtained if the total surface area of the cube is known to us. This can be done by rearranging the formula for the total surface area of the cube.

Total surface area (TSA) of cube = 6a2 sq. units

Or,    TSA = 6 × (Side)2 = 6 × (Edge)2

Or,    (Edge)2 = \[\frac{{TSA}}{6}\]

Or,    Length of edge of cube = \[\sqrt {\frac{{TSA}}{6}} \]  units

Lateral Surface Area (LSA) of Cube

Lateral surface area of the cube is the area of its all faces excluding the area of top and bottom faces. 

Therefore, the Lateral surface area of cube = 4 × (side)2 sq. units

Or, Lateral surface area (LSA) of cube = 4a2 sq. units

Solved Examples:

Q1. Find the Area of Metal Sheet Required to Make a Cubical Box of Side Length 5m.

Solution: Given, length of side of cubical box = 5m.

The area of metal sheet required to make a cubical box = TSA of cube 

                                                                                         = 6 × (Side)2

                                                                                                                                  = 6 × (5)2 sq. mts

                                                                                         = 6 × 25 sq. mts

                                                                                         = 150 sq. mts  

Hence, the area of metal sheet required to make a cubical box of side length 5m is 150 sq. mts

Q2. Find the Ratio of the Lateral Surface Area and Total Surface Area of a Cube.

Solution: Let the length of the side of the cube be a unit. Then,

Lateral surface area (LSA) of cube = 4 a 2
Total surface area (TSA) of cube = 6 a 2

Ratio of LSA and TSA of cube = \[\frac{{LSA\,of\,cube}}{{TSA\,of\,cube}}\]

                                                  \[ = \frac{{4{a^2}}}{{6{a^2}}} = \frac{2}{3}\]
Hence, the ratio of the lateral surface area and total surface area of a cube is 2 : 3

Q3. Find the Length of the Edge of the Cube whose Surface Area is 600 cm2.

Solution:  Given, Surface area of cube = 600 cm             

We know, Length of edge of cube = \[\sqrt {\frac{{TSA}}{6}} \]  units

⇒ Length of edge of cube = \[\sqrt {\frac{{600}}{6}} \]cms

                                           = \[\sqrt {100} \]= 10cms

Q4. A Cubical Container of Side 4m is to be Painted on the Entire Outer Surface. Find the Area to be Painted and the Total cost of Painting it at the Rate of ₹24 per m2.

Solution: Given, Side length of cubical container = 4 m

Since the container is to be painted on the entire outer surface, the area to be painted can be obtained by finding the total surface area of the cubical container.

Therefore, total surface area of cubical container = 6 × (Side)2

                                                                                                               = 6 × (4)2 sq. mts

                                                                            = 6 × 16 sq. mts

                                                                            = 96 sq. mts

Given, cost of painting 1 sq. mtr is ₹24

So, the total cost of painting the cubical container = ₹ (96 × 24) = ₹ 2304