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Surface Area of Cuboid

The surface area of a cuboid is the total space occupied by it. The cuboid is a six-faced three-dimensional shape whose each face resembles a rectangle. We will learn to deduce the formula of the surface area of the cuboid and learn to apply the formula as well. Once you understand this chapter you will learn to solve problems on the surface area of the cuboid.

1.	What is the Surface Area of Cuboid?
2.	Surface Area of Cuboid Formula
3.	Derivation of Surface Area of Cuboid
4.	How to Calculate the Surface Area of Cuboid?
5.	FAQs on Surface Area of Cuboid

What is the Surface Area of Cuboid?

The surface area of the cuboid is the total area of all its surfaces. As, a cuboid is a three-dimensional (dimensions being length, breadth, and height) solid shape the value of its surface area will depend on the dimensions of length, breadth, and height. The change in any of the dimensions of a cuboid changes the value of the surface area of a cuboid. The unit of the surface area of a cuboid is given as the (unit)². The metric units of surface area are square meters or square centimeters while the USCS units of surface area are, square inches or square feet.

Surface Area of Cuboid Formula

A cuboid can have two kinds of surface areas:

Total Surface Area
Lateral Surface Area

The total surface area of the cuboid is obtained by adding the area of each face while the lateral surface area of the cuboid is found by finding the area of each face excluding the base and the top. The total surface area is denoted by "S" while the lateral surface area is denoted by "L". The total surface area and lateral surface area can be expressed in terms of length (l), breadth(b), and height of cuboid(h) as:

Total Surface Area of Cuboid, S = 2 (lb + bh + lh)
Lateral Surface Area of Cuboid, L = 2h (l + b)

We can derive the formulas for surface area of cuboid can be derived by opening up a cuboidal box. Let us see how.

Derivation of Surface Area of Cuboid

Let us say a cuboid of dimensions (l × b × h) having 6 faces numbered as 1, 2, 3, 4, 5, and 6 is opened. We have marked the six faces with numbers.

Cuboid Surface Area Deduction

Total Surface Area of Cuboid

The total surface area of the six faces is given by adding all the areas of each face. Let the total surface area of the cuboid be "S". As each face of a cuboid is a rectangle, hence the area of the rectangle is found for each face and added in the end to determine the total surface area of the cuboid.

Face Number	Area of Rectangle
1	l × b
2	l × b
3	l × h
4	l × h
5	b × h
6	b × h

Hence, the total surface area of six faces = (l × b) + (l × b) + (l × h) + (l × h) + (b × h) + (b × h).
Clearly, total surface area, S = (2 × l × b) + (2 × b × h) + (2 × l × h)
Thus, the surface area S of any cuboid of dimensions l, b, and h will be S = 2 (lb + bh + lh)

Lateral Surface Area

The lateral surface area of a cuboid is the combined surface area of the four vertical faces. In the diagram above, the two horizontal faces will be removed from the total surface area and only the vertical faces remain, that is face 1 and face 2. The lateral surface area of cuboid will be, L= S = 2 (lb + bh + lh) - (2 × l × b) = (2 × l × h) + (2 × b × h) = 2h(l + b).

In order to understand better let us assume there is a cuboidal room. The total surface area will be the combined area of the six faces of the room (the four vertical walls + the floor + the ceiling) while the lateral surface area will be the combined surface area of the four vertical walls (the areas of the floor and the ceiling are not added).

How to Calculate Surface Area of Cuboid?

The surface area of the cuboid is the area of each surface of a cuboid. The surface of a cuboid can be calculated by understanding the need of the kind of area in a particular situation. The steps to calculate the surface area of a cuboid are:

Step 1: Check if the given dimensions of cuboids are in the same units or not. If not, convert the dimensions into the same units.
Step 2: Once the dimensions are in the same units, understand the need to calculate the total surface area or lateral surface area in a particular situation.
Step 3: Implement the formula for total surface area, 2 (lb + bh + lh) or lateral surface area 2h(l +b).
Step 4: Write the unit with the value obtained.

Let us take an example to learn how to calculate the surface area of a cuboid using its formula.

Example: Find the total surface area and lateral surface area of a cuboid having length 4 inches, breadth 6 inches and height 3 inches.
Solution: As we know, the total surface area of cuboid, S = 2 (lb + bh + lh) and lateral surface area of a cuboid is 2h(l + b).
Here, length l = 4 inches, breadth b = 6 inches and height h = 3 inches

Thus, total surface area of cuboid, S = 2 (lb + bh + lh) = 2 ((4 × 6) + (6 × 3) + (4 × 3)) in²
⇒ S = 108 in²
Thus, lateral surface area of cuboid, L = 2h(l + b) = (2 × 3)(4 + 6) in²
⇒ L = 60 in²
\(\therefore\) The total surface area of cuboid and lateral surface area of a cuboid is 108 in² and 60 in² respectively.

Example 1: What is the height of the cuboidal room, if its lateral surface area is 900 in² and the dimensions of breadth and length are, 10 inches and 20 inches?
Solution: As we know, the lateral surface area of a cuboid is L = 2h(l + b)
Length = 20 in
Breadth = 10 in
Let Height = h in

The lateral surface area of cuboid is given as:
L = 2h(l +b)
⇒ L = 2 h(20 + 10) in²= 900 in²
⇒ h = 900/(2 × 30) = 15 in
\(\therefore\) The height of the cuboidal room is 15 inches.
Example 2: What will be the length of the cuboid if its total surface area is 1600 unit²while the breadth and height of the cuboid are double the value of length?
Solution: As we know, the surface area of a cuboid is given as S = 2 (lb + bh + lh). The given dimensions for cuboid are:
Total Surface Area = 1600 unit²
Let the length of the cuboid is l units.
Breadth = b = 2 × length = 2l units
Height = h= 2 × length = 2l units

Hence, the total surface area of the cuboid will be:
S = 2 (lb + bh + lh)
⇒ S = 2 ((l × 2l) + (2l × 2l) + (l × 2l)) unit² = 1600 unit²
⇒ S = 2 (2l² + 4l² + 2l²) unit²= 2(8l²) = 1600 unit²
⇒ 16 l² = 1600 unit²
⇒ l² = 100 unit²
⇒ l = 10 units
\(\therefore\) The length of cuboid is 10 units.

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FAQs on Surface Area of Cuboid

What is the Total Surface Area of Cuboid?

The total surface area of a cuboid is the sum of all its surfaces. The formula for the total surface area of a cuboid is given as, 2 (lb + bh + lh) where l, b, and h indicate, length, breadth, and height of the cuboid.

What is the Lateral Surface Area of a Cuboid With Dimensions, L, B, and H?

The lateral surface area of a cuboid is the value of the surface area of a cuboid excluding its horizontal surfaces. The formula for the lateral surface area of a cuboid is given as, 2H(L + B) where L, B, and H indicate the dimensions of length, breadth, and height of the cuboid.

What is the Difference Between the Total Surface Area and the Lateral Surface Area of a Cuboid?

The difference between total surface area and the lateral surface area of a cuboid is:

The total surface area of a cuboid is the sum of areas of each face while the lateral surface area of a cuboid is the sum of areas of faces excluding the top and the base.
The total surface area of a cuboid is found using formula 2 (lb + bh + lh) while the lateral surface area of a cuboid is found using formula 2h(l + b).

What is the TSA and LSA of a Cuboid?

The TSA and LSA of a cuboid are the abbreviations of Total Surface Area and Lateral Surface Area of a cuboid. The total surface area is the sum of the surface area of all faces while lateral surface area is the measure of surface area excluding the two horizontal faces.

What Will Be the Total Surface Area of a Cuboidal Box Without Lid?

The total surface area of a cuboidal box without a lid can be given in two ways:

Way 1: Total surface area of a Cuboidal Box Without Lid = Total surface area of the cuboid - 1 rectangular face;
Way 2: Total surface area of a Cuboidal Box Without Lid = Lateral surface area of cuboid + 1 rectangular face;

Both of them result in the same answer.

What is the Relation Between Volume and Surface Area of a Cuboid?

The volume of a cuboid can be expressed as the product of the height and area of one surface of the cuboid.

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