Surface Area of a Rectangular Prism

A rectangular prism is a three-dimensional geometric figure that has four lateral faces with two congruent and parallel bases. A rectangular prism is a polyhedron, and every face is a rectangle. A rectangular prism has a total of six faces where the opposite faces are identical, i.e., a rectangular prism has three pairs of identical faces. The dimensions of a rectangular prism are length, width, and height. It has a total of six faces, twelve edges, and eight vertices. Some examples of rectangular prisms that we see in our everyday lives are fish tanks, notebooks, diaries, cargo containers, rooms, etc.

Surface Area of a Rectangular Prism

The total area occupied by all the three-dimensional surfaces of a three-dimensional geometric structure is called the surface area. The surface area of the prism is equal to the area of its net. So, to find the surface area of a rectangular prism, we have to calculate the areas of each of its faces, then add the resulting areas. A rectangular prism has two types of surface areas: a lateral surface area and a total surface area.

Surface Area of Rectangular Prism Formula

There are two formulas that are used to calculate the area of a rectangular prism which are,

Lateral Surface Area Formula

The lateral surface area of a prism (LSA) is equal to the sum of the areas of its four lateral faces.

Lateral Surface Area of a Prism (LSA) = Sum of areas of four lateral faces.

So, the formula for calculating the lateral surface area of a rectangular prism is given as follows:

Lateral Surface Area of a Rectangular Prism = 2h (l + b) square units

where,
“l” is the length of the side of a base,
“b” is the breadth of the side of a base,
“h” is the height of the prism.

Total Surface Area Formula

The total surface area of a rectangular prism is equal to the sum of the total areas of all its faces. 

Total Surface Area of a Prism (TSA) = LSA + 2 × Base area

So, the formula for calculating the total surface area of a rectangular prism is given as follows:

TSA = 2(lb + bh + lh) square units

where,
“l” is the length of the side of a base,
“b” is the breadth of the side of a base,
“h” is the height of the prism.

How to Find the Surface Area of a Rectangular Prism?

Let us go through an example to understand the concept of calculating the surface area of a rectangular prism.

Example: Calculate the surface area of a rectangular prism if its height is 15 units and the length and breadth of the base are 10 units and 6 units, respectively.

Step 1: Note the dimensions of the given rectangular prism. In the given example, the length and breadth of the rectangular prism’s base are 10 units and 6 units, respectively, and its height is 15 units.

Step 2: We know that the surface area of a rectangular prism is equal to 2(lb + bh + lh) square units. Now, substitute the given values of length, breadth, and height in the formula.

Step 3: So, the surface area of the rectangular prism is calculated as, A = 2× (10 × 6 + 6 × 15 + 10 × 15) = 600 sq. units.

Using the above steps Surface Area of a Rectangular Prism is found.

Solved Problems on the Surface Area of Rectangular Prism

Problem 1: Determine the total surface area of a rectangular prism if its lateral surface area is 560 sq. cm and the length and breadth of the base are 12 cm and 8 cm, respectively.

Solution:

Given data,

length of the rectangular base (l) = 12 cm

The breadth of the rectangular base (b) = 8 cm

The lateral surface area of the prism (LSA) = 560 sq. cm

We have,

The total surface area of a prism (TSA) = LSA + 2 × Base area

Base area = 2(l + b)

= 2 × (12 + 8) = 2 × 20 = 40 sq. cm

Now, TSA = 560 + 2 × 40

= 560 + 80 = 640 sq. cm

Hence, the rectangular prism’s total surface area is 640 sq. cm.

Problem 2: Calculate the length of the base of a rectangular prism if its height is 9 inches and the breadth of the base is 4 inches, and the lateral surface area is 198 sq. in.

Solution:

Given data,

The lateral surface area = 198 sq. in

The breadth of the rectangular base (b) = 4 inches

Height = 9 inches

length of the rectangular base (l) =?

We have,

The Lateral Surface Area of a Rectangular Prism = 2h (l + b) square units

⇒ 2 × 9 × (l + 4) = 198

⇒ 18 × (l + 4) = 198

⇒ l + 4 = 198/18 = 11

⇒ l = 11 − 4 = 7 in

Thus, the length of the rectangular prism is 7 inches.

Problem 3: Find the lateral surface area of a rectangular prism if its height is 18 cm and the length and breadth of the base are 14 cm and 10 cm, respectively.

Solution:

Given data,

The length of the rectangular base (l) = 14 cm

The breadth of the rectangular base (b) = 10 cm

Height = 18 cm

We know that,

The Lateral Surface Area of a Rectangular Prism = 2h (l + b) square units

= 2 × 18 × (14 + 10)

= 36 × 24 = 864 sq. cm

Hence, the lateral surface of the given prism is 864 sq. cm.

Problem 4: Determine the surface area of a rectangular prism if its height is 12 cm and the length and breadth of the base are 8 cm and 5 cm, respectively.

Solution:

Given data,

The length of the rectangular base (l) = 8 cm

The breadth of the rectangular base (b) = 5 cm

Height = 12 cm

We have,

The Total Surface Area of a Rectangular Prism = 2(lb + bh + lh) square units

= 2 × (8 × 5 + 5 × 12 + 8 × 12)

= 2 × (40 + 60 + 96)

= 2 × 196 = 392 square units

Hence, the rectangular prism’s surface area is 392 square units.

Problem 5: Determine the surface area of a rectangular prism if its height is 14 units and the length and breadth of the base are 10 units and 7 units, respectively.

Solution:

Given data,

The length of the rectangular base (l) = 10 units

The breadth of the rectangular base (b) = 7 units

Height = 14 units

We have,

The Total Surface Area of a Rectangular Prism = 2(lb + bh + lh) square units

= 2 × (10 × 7 + 7 × 14 + 10 × 14)

= 2 × (70 + 98 + 140)

= 2 × 308 = 616 square units

Hence, the rectangular prism’s total surface area is 616 square units.

FAQs on Rectangular Prism

Question 1: What is meant by a rectangular prism?

Answer:

In mathematics, a rectangular prism is a three-dimensional geometric figure that has four lateral faces with two congruent and parallel bases. The dimensions of a rectangular prism are length, width, and height. It has a total of six faces, twelve edges, and eight vertices.

Question 2: Mention some examples of a rectangular prism.

Answer:

Some examples of rectangular prisms that we see in our everyday lives are fish tanks, notebooks, diaries, cargo containers, rooms, etc. 

Question 3: What is the total surface area of a rectangular prism?

Answer:

The total surface area of a rectangular prism is equal to the sum of the total areas of all its faces. 

TSA = 2(lb + lh + bh) square units

Where “l” is the length of the side of a base, “b” is the breadth of the side of a base, and “h” is the height of the prism.

Question 4: What is the lateral surface area of a rectangular prism?

Answer:

The lateral surface area of a prism (LSA) is equal to the sum of the areas of its four lateral faces.

So, the formula for calculating the lateral surface area of a rectangular prism is given as follows:

LSA = 2h (l + b) square units

Where “l” is the length of the side of a base, “b” is the breadth of the side of a base, and “h” is the height of the prism.