# Surface Area of a Cuboid – Explanation & Examples

Before we get started, let’s discuss what a cuboid is. A cuboid is one of the most common shapes in the environment around us. For example, a brick, a matchbox, a chalk box, etc., are all cuboids.

In geometry, a cuboid is a 3-dimensional figure with a length, width, and height. A cuboid has 6 rectangular faces. Ultimately, a cuboid has the shape of a rectangular prism or a box.

In a cuboid, the horizontal longer side is the length (l), and the shorter horizontal side is the width (w) or breadth (b). The height (h) of a cuboid is the vertical side.

The surface area of a cuboid is the sum of the area of the 6 rectangular faces that cover it.

In this article, we will learn how to find the surface area using a cuboid formula’s surface area.

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

To find the surface area of a cuboid, you need to calculate the area of each rectangular face and then sum up all the areas to get the total surface area i.e.

• Area of the top and bottom face = lw+ lw = 2lw
• Area of the front and back face = lh+ lh = 2lh
• Area of the two side faces = wh+ wh = 2wh

The total surface area of a cuboid is equal to the sum of the face areas;

Surface area of cuboid = 2lw + 2lh + 2wh

Note: The cuboid’s total surface area is not the same as the lateral surface area of a cuboid. The lateral surface of a cuboid is the sum of the area of the rectangular faces excluding the top and bottom face;

Lateral surface area of a cuboid (LSA) = 2h (l +b)

### Surface area of a cuboid formula

From the above illustration, the formula for the total surface area of a cuboid can be represented as:

Total surface area of a cuboid (TSA) = 2 (lw + wh + lh)

The units for the surface area of a cuboid are square units.

Let’s practice some example problems below.

Example 1

The dimensions of a cuboid are given as follows:

Length = 5 cm

Width = 3 cm

Height = 4 cm.

Find the total surface area of the cuboid.

Solution

By the formula,

Total surface area of a cuboid = 2 (lw + wh + lh)

Substitute.

TSA = 2(5 x 3 + 3 x 4 + 5 x 4)

= 2(15 + 12 + 20)

= 2(47)

= 2 x 47 = 94 cm2

Therefore, the total surface area of the cuboid is 94 cm2

Example 2

The surface area of a cuboid is 126 ft2. If the cuboid’s length and height are 6 feet and 3 feet, find the width of the cuboid.

Solution

Given;

Total surface area = 126 ft2

Length = 6 ft

Height = 3 ft

Therefore,

⇒126 = 2 (lw + wh + lh)

⇒126 = 2 (6w + 3w + 6 x 3)

⇒126 = 2(9w + 18)

⇒126 = 18 w + 36

Subtract by 36 on both sides and then divide by 18

90 = 18 w

w = 5

Therefore, the width of the cuboid is 5 feet.

Example 3

Given the dimensions of a cuboid as:

Length = 10 m

width = 5 width

Height = 9 m

By how much is the total surface area of the cuboid more than the lateral surface area?

Solution

Total surface area = 2 (lw + wh + lh)

= 2 (10 x 5 + 5 x 9 + 10 x 9)

= 2(50 + 45 + 90)

TSA = 2 x 185

=370 m2.

The lateral surface area of a cuboid = 2h (l + b)

= 2 x 9(10 + 5)

= 18 x 15

= 270 m2

Total surface area – lateral surface area = 370 – 270

= 100 m2

Therefore, the total surface area of the cuboid is 100 m2 more than the lateral surface area.

Example 4

The length and width of a cardboard are 20 m by 10 m, respectively. How many cuboids can be made from the cardboard if each cuboid must be 4 m long, 3 m wide, and 1 m high.

Solution

Area of the cardboard = l x w

= 20 x 10

= 200 m2

Total surface area of the cuboid = 2 (lw + wh + lh)

= 2 (4 x 3 + 3 x 1 + 4 x 1)

= 2 (12 + 3 + 4)

= 2 x 19

= 38 m2

The number of cuboids = area of the cardboard/total surface area of a cuboid

= 200 m/38 m2

= 5 cuboids

Example 5

Compare the total surface area of a cube of length 8 cm and a cuboid of length 8 cm, width, 3 cm, and height, 4 cm.

Solution

Total surface area of a cube = 6a2

= 6 x 82

= 6 x 64

= 384 cm2

Total surface area of a cuboid = 2 (lw + wh + lh)

= 2(8 x 3 + 3 x 4 + 8 x 4)

= 2(24 +12 + 32)

= 2 x 68

= 136 cm2

Therefore, the surface area of the cube is more than the surface area of the cuboid.

### Practice Questions

1. The dimensions of a cuboid are given as follows:

Length = $12$ ft

Width = $6$ ft

Height = $4$ ft

What is the total surface area of the cuboid?

2. The surface area of a cuboid is $664$ m2. If the cuboid’s length and height are $8$ meters and $10$ meters, what is the width of the cuboid?

3. The dimensions of a cuboid are given as follows:

Length = $20$ m

Width = $4$ m

Height = $12$ m

What is the total surface area of the cuboid?

4. The length and width of the given cardboard are $12$ m by $15$ m long, respectively. How many cuboids can be made from the cardboard if each cuboid must be $2$ m long, $1.5$ m wide, and $0.5$ m high?

5. Which of the following has the largest total surface area?