The sum of the areas of all faces (excluding top and bottom) of a cuboid is the _ of the cuboid.

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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.

The surface area of a cuboid is the total space occupied by it. A cuboid is a six-faced three-dimensional shape in which each face is in the shape of a rectangle. Let us learn more about the formula of the surface area of the cuboid and solve problems based on the surface area of the cuboid.

What is the Surface Area of Cuboid?

The surface area of a cuboid is the total area of all its surfaces. Since a cuboid is a three-dimensional solid shape, the value of its surface area depends on the dimensions of its length, width, and height. The change in any of the dimensions of a cuboid changes the value of the surface area of a cuboid. The surface area of a cuboid is expressed in square units.

Surface Area of Cuboid Formula

The formula for the surface area of a cuboid depends on the type of surface area that has been asked for. A cuboid has 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 all the 6 faces while the lateral surface area of the cuboid is calculated by finding the area of each face excluding the base and the top. The total surface area and the lateral surface area of a cuboid can be expressed in terms of length (l), width (w), and height of cuboid (h) as:

Total Surface Area of Cuboid = 2 (lw + wh + lh)
Lateral Surface Area of Cuboid = 2h (l + w)

If the surface area is given or asked without any specifications, it means that it refers to the total surface area. Let us see how to derive the formulas for the surface area of the cuboid by opening up a cuboidal box.

Derivation of Surface Area of Cuboid

A cuboid has 6 rectangular faces. Now, in order to find the total surface area of a cuboid, we need to add the area of all the 6 rectangular faces.

Total Surface Area of Cuboid

The total surface area of the six faces can be calculated by adding the areas of each face. Since each face of a cuboid is a rectangle, hence the area of the rectangle for each face is calculated and added to get the total surface area of the cuboid. Let us understand this with the help of the figure given below.

All the faces are numbered as 1, 2, 3, 4, 5, and 6 as shown in the figure given above. In other words, if we see the cuboid in the form of a two-dimensional figure as a net, we get this figure.

• Area of Rectangle 1 and 2: This is the area of the rectangle of the top and bottom faces = l × w. So, this makes it, lw + lw = 2lw
• Area of Rectangle 3 and 4: This is the area of the rectangle of the front and back faces = w × h. So, this makes it, wh + wh = 2wh
• Area of Rectangle 5 and 6: This is the area of the rectangle of the faces on the left and right side = l × h. So, this makes it, lh + lh = 2lh

Hence, the total surface area of the six faces = 2lw + 2wh + 2lh
Thus, the total surface area of a cuboid of dimensions l, w, and h will be = 2 (lw + wh + lh)

Lateral Surface Area of Cuboid

The lateral surface area of a cuboid is the combined surface area of the four vertical faces. In the figure given above, if we remove the top and bottom faces, we will get the area of the lateral surface area of the cuboid. The lateral surface area of the cuboid is,

Lateral Surface Area = Total Surface Area - Area of top and bottom faces

Lateral Surface Area = 2 (lw + wh + lh) - (2 × l × w)

= 2lw + 2wh + 2lh - 2lw

= 2lh + 2wh

= 2h(l + w)

To understand this with the help of a real-life example let us imagine a room in the shape of a cuboid. 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 Find the Surface Area of Cuboid?

The surface area of a cuboid is the total area of each surface of a cuboid. Let us use the following steps to calculate the total surface area of a cuboid.

• 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: Use the formula for total surface area = 2(lw + wh + lh)
• Step 3: Substitute the given values to get the area and express it in square units.

Let us learn how to calculate the total surface area and the lateral surface area of a cuboid with the help of an example.

Example: Find the total surface area and the lateral surface area of a cuboid whose length = 6 inches, width = 4 inches, and height = 3 inches.

Solution: As we know, the total surface area of a cuboid = 2 (lw + wh + lh) and the lateral surface area of a cuboid is 2h(l + w).

Here, length (l) = 6 inches, width (w) = 4 inches and height (h) = 3 inches

Total surface area of cuboid = 2 (lw + wh + lh) = 2 [(6 × 4) + (4 × 3) + (6 × 3)] in2

⇒ Total surface area of cuboid = 108 in2

Lateral surface area of cuboid = 2h(l + w) = (2 × 3)(6 + 4) in2

⇒ Lateral surface area of cuboid = 60 in2

Therefore, the total surface area of the cuboid is 108 in2 and the lateral surface area of the cuboid is 60 in2.

☛ Related Articles

1. Example 1: Find the lateral surface area of a cuboid whose length = 20 inches, width = 10 inches, and height = 15 inches.

   Solution: As we know, the lateral surface area of a cuboid is L = 2h(l + w) and it is given that,

   length (l) = 20 in

   width (w) = 10 in

   height (h) = 15 in

   Let us substitute the values in the formula,

   Lateral surface area of cuboid = 2h(l + w)
   ⇒ 2 × 15(20 + 10)

   ⇒ 30(20 + 10) ⇒ 30 × (30) = 900 square inches

   Therefore, the lateral surface area of the cuboid is 900 square inches.

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FAQs on Surface Area of Cuboid

The total surface area of a cuboid is the sum of all its surfaces. In order to find the total surface area of a cuboid, we need to add the area of all the 6 rectangular faces. The formula for the total surface area of a cuboid is 2 (lw + wh + lh) where l = length, w = width, and h = height of the cuboid.

What is the Lateral Surface Area of a Cuboid?

The lateral surface area of a cuboid is the value of the surface area of a cuboid excluding its top and bottom surfaces. The formula for the lateral surface area of a cuboid is expressed as, 2h(l + w) where l = length, w = width, and h = 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 given below:

• The total surface area of a cuboid is the sum of the areas of all 6 faces, whereas, the lateral surface area of a cuboid is the sum of the areas of faces excluding the top and the base.
• The total surface area of a cuboid is calculated using formula 2(lw + wh + lh), whereas, the lateral surface area of a cuboid is calculated using formula 2h(l +w).

What is the TSA and LSA of a Cuboid?

The TSA and LSA of a cuboid are the abbreviations of Total Surface Area (TSA) and Lateral Surface Area(LSA) of a cuboid. The total surface area is the sum of the surface area of all 6 faces while lateral surface area is the measure of surface area excluding the top and bottom faces.

What is 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:

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

It should be noted that both the methods result in the same answer.

What is the Unit for Surface Area of Cuboid?

The surface area of a cuboid is expressed in square units like cm2, m2, inch2, and so on.

What is the Formula to Find the Total Surface Area of a Cuboid?

The formula that is used to find the total surface area of a cuboid is 2 (lw + wh + lh); where l = length, w = width, and h = height of the cuboid.

What is the Formula to Find the Lateral Surface Area of a Cuboid?

The formula that is used to find the lateral surface area of a cuboid is 2h(l + w) where l = length, w = width, and h = height of the cuboid.