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# Surface Area of a cube – Explanation & Examples

**Finding the surface area of an object is important if you want to determine how much material is needed to cover an object’s surface.**

For example, companies that package items in carton boxes require the surface area to determine how much cardboard would be needed to make the box.

**The surface area of a cube is the total sum of the area of all the six squares that cover a square.**

In this article, we will learn how to find the surface area of a cube using the surface area of a cube formula.

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

To recall, a cube is a 3-dimensional figure with 6 equal square faces, 8 edges, and 8 vertices. Since a cube has six faces, a cube’s surface area is found by multiplying the area of one square face by 6.

As for other areas, an object’s surface area is measured in square units, i.e., mm^{2}, cm^{2}, m^{2}.

### Surface area of a cube formula

From the above illustration, the surface area of a cube is equal to:

Surface area of a cube = a^{2} + a^{2} + a^{2} + a^{2} + a^{2} + a^{2}

Therefore, the surface area of a cube formula is given as:

**Surface area of a cube = 6a ^{2}**

where a = any side length of a cube.

Let’s work out some example problems involving the surface area of a cube.

*Example 1*

Find the surface area of a cube of a side length of 10 cm.

__Solution__

By the formula,

Surface area of a cube = 6a^{2}

**= **6 x 10^{2}

= 6 x 100

= 600 cm^{2}

*Example 2*

Find the surface of a cube whose volume is 343 m^{3}.

__Solution__

Given

Volume of a cube, a^{3 }= 343 m^{3}

First find the length of the cube

a = ^{3}√343

a = 7 m

SA = 6a^{2}

= 6 x 7^{2}

= 6 x 49

= 294 m^{2}

*Example 3*

The surface area of a cube is 150 feet square. What is the length of the cube?

__Solution__

Given, surface area = 150 ft^{2}

SA = 6a^{2}

150 = 6a^{2}

Divide both sides by 6 to get,

25 = a^{2}

√a = 5

Therefore, the length of the cube is 5 feet.

*Example 4*

A solid cube of length 10 m is to be painted on its 6 faces. If the painting rate is $ 10 per square meter, find the total cost of painting the cube.

__Solution__

To find the total cost of painting a cube, we multiply the cube’s surface area by the rate of painting.

SA = 6a^{2}

= 6 x 10^{2}

= 6 x 100

= 600 m^{2}

The cost of painting = 600 m^{2} x $ 10 per m^{2}

= $6000.

*Example 5*

The height of a cubical tank is 12 feet. Find the surface area of the tank.

__Solution__

SA = 6a^{2}

= 6 x 12^{2}

= 6 x 144

= 864 ft^{2}

*Example 6*

What is the length of the side of a cube whose surface area is equal to its volume?

__Solution__

Given:

Surface area of a cube = volume of a cube

6a^{2 }= a^{3}

Divide both sides by a^{2}

6a^{2}/a^{2} = a^{3}/a^{2}

6 = a

Therefore, the length of the cube is 6 units.

*Example 7*

Find the surface area of a cube whose diagonal is 12 yards.

__Solution__

For a cube, the length of the diagonal = **√**3a

where a = side length of a cube.

Therefore,

12 = **√**3a

Square both sides and then divide by 3.

144 = 3a

a = 48

Now, calculate the surface area of the cube

SA = 6a^{2}

= 6 x 48 x 48

= 13824 square yards

*Example 8*

A rectangular cardboard is 0. 5 m long and 0.3 m wide. How many cubical boxes of length 5 cm can be made from cardboard?

__Solution__

The area of the rectangular cardboard = 0.5 x 0.3

= 0.15 m^{2} ⇒ 1,500 cm^{2}

Surface area of a cubical box = 6a^{2}

= 6 x 5^{2}

= 6 x 25

= 150 cm^{2}

To get the number of boxes, divide the area of the card by the surface area of a cube

Number of boxes = 1,500/150

= 10 boxes.

*Example 9*

The cost of 1 m^{2} of a card is $ 0.5. Find the cost of making 60 cubical boxes of length 0. 4 m.

__Solution__

First, determine the surface area of the 60 boxes

SA of a box = 6a^{2}

= 6 x 0.4^{2}

= 6 x 0.16

= 0.96 m^{2}

Surface area of 60 boxes = 0.96 x 60

= 57.6 m^{2}

The cost of making 60 boxes = 57.6 x 0.5

= $28.8

*Example 10*

The surface area of a cube is 1014 in^{2}. What is the volume of the cube?

__Solution__

SA = 6a^{2}

1014 = 6a^{2}

a^{2} = 169

a = √169

a =13

The volume of a cube = a^{3}

= 13 x 13 x 13

= 2197 in^{3}.