Surface Area of a Rectangular Prism Formula

In geometry, a rectangular prism is a polyhedron with two congruent and parallel bases. It is also called a cuboid. A rectangular prism has six faces, and all the faces are in a rectangle shape and have twelve edges. Because of its cross-section along the length, it is said to be a prism. Geometry is the study of shapes and the configuration of objects. Similar to other three-dimensional shapes, a rectangular prism also has its surface area and volume. The surface area of the prism is the area of its net. In this article, let us discuss the definition, types, surface area, and volume of a rectangular prism in detail.

What is a Rectangular Prism?

A rectangular prism is a three-dimensional shape, that has six faces (two at the top and bottom and four are lateral faces). All the faces of the prism are rectangular in shape. Hence, there are three pairs of identical faces here. Due to its shape, a rectangular prism is also called a cuboid. Some of the real-life examples of a rectangular prism are a geometry box, notebooks, diaries, rooms, etc. In the below figure, we can see the shape of a rectangular prism.

Properties of Rectangular Prism

• A rectangular prism has 6 faces, 12 edges and 8 vertices
• The top and base of the rectangular prism are always a rectangle
• Like cuboid, it also has three dimensions, i.e., length width and height
• Pairs of opposite faces are identical or congruent
• For a right rectangular prism, the lateral faces are rectangle
• For an oblique rectangular prims, the lateral faces are parallelogram
• It has a rectangular cross-section
• It looks exactly like a cuboid

Types of Rectangular Prism

Rectangular prism can be classified into two different types. They are:

1. Right Rectangular Prism
2. Oblique Rectangular Prism

Right Rectangular Prism

A prism with rectangular bases is called a rectangular prism. A right rectangular prism is a prism that has six faces that are rectangles, and all angles are right angles.

• Vertices of a rectangular prism = 8
• Edges of a rectangular prism = 12
• Faces of a rectangular prism= 6 (including bases)

Oblique Rectangular prism

An oblique prism is a prism in which the bases are not perpendicular to each other. A rectangular prism with bases that are not aligned one directly above the other is an oblique rectangular prism.

Rectangular Prism Formulas

A rectangular prism is a three-dimensional object. Hence, it will have its surface area and volume. To calculate the volume and surface area of a prism, we have to know the length of its sides or edges. Let ‘l’, ‘w’ and ‘h’ be the length, width and height of the rectangular prism. The formulas are given below.

Volume of a Rectangular Prism  Formula

The volume of a rectangular prism is a measurement of the occupied units of a rectangular prism. The volume of a rectangular prism is represented by cubic units. It is also defined as the number of units used to fill a rectangular prism.

The volume of the rectangular prism is equal to the area of the base times its height.

Therefore, the volume of a rectangular prism formula is given as

The volume of a rectangular prism = Length x Width x Height cubic units.

Surface Area of a Rectangular Prism

The surface area of a rectangular prism is the measure of how much-exposed area a prism has. Surface area is expressed in square units. The total surface area of a rectangular prism is the sum of the lateral surface area (LSA) and twice the base area of the rectangular prism.

Total Surface Area of rectangular prism = LSA + 2 (Base area)       [Square units]

The lateral surface area of a rectangular prism is the sum of the surface area of all its faces without the base of the rectangular prism. The lateral surface area of any right rectangular prism is equivalent to the perimeter of the base times the height of the prism.

Therefore, the lateral surface area = P x h [Square units]

Where

P is the perimeter of a base

h be the height of the prism

The perimeter of the rectangular prism is,

Therefore, the lateral surface area (LSA) of a rectangular prism = 2 ( l + w ) h square units.

Hence,

TSA = LSA + 2 (Base Area) = 2 (l + w) h + 2 (l x w) = 2 lh + 2 wh + 2 lw   [Square units]

Therefore, the surface area of a rectangular prism formula is given as,

Rectangular Prism Net

The net of any prism is its surface area. It shows when we open the prism in a plane; then all its sides could be visible at the same time. If we calculate the individual area of all its sides using the net, we will get the total surface area. See the figure below to find the net for the rectangular prism.

 

You can see from the above figure, all the sides of the prism are in a rectangular shape. By using the formula for the area of rectangle, you can find the areas for each face and add all the areas to get the net of the prism.

Solved Examples on Rectangular Prism

Question 1: Find the volume of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively.

Solution:

Given:

Length, l = 8 cm

Width, w = 6 cm

Height, h = 4 cm

The formula to find the volume of a rectangular prism is,

V = Length x Width x Height cubic units

V = 8 x 6 x 4 cm³

V = 192 cm³

Therefore, the volume of a rectangular prism is 192 cm³.

Question 2: Find the surface area of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively.

Solution: Given:

Length, l = 8 cm

Width, w = 6 cm

Height, h = 4 cm

The formula to find the area of a rectangular prism is,

A = 2 (lh +wh + lw )

A = 2 (8×4+6×4+8×6)

A = 2(32+24+48)

A = 2(104)

A = 208 sq.cm.

Practice Questions

1. Find the area of a rectangular prism whose length, width, and height are given, respectively.

• 3cm, 4cm and 5cm
• 2.5 cm, 6cm, 9cm
• 5 cm, 8cm, 10cm
• 6.2 cm, 4.4 cm, 9cm

2. Find the volume of rectangular prism with the following dimensions.

• 3cm x 4cm x 5cm
• 2.5 cm x 6cm x 9cm
• 5 cm x 8cm x 10cm
• 6.2 cm x 4.4 cm x 9cm

Surface Area of Rectangular Prism

The surface area of a rectangular prism is the total area or region covered by its six faces. Prisms are solids with flat parallelogram sides and identical polygon bases. There are different types of prisms, namely – triangular prisms, square prisms, rectangular prisms, pentagonal prisms, hexagonal prisms, etc. In this article, you will learn more about the surface area of rectangular prisms.

The total region or area covered by all the faces of a rectangular prism is defined as the surface area of a rectangular prism. A rectangular prism is a three-dimensional shape. It has six faces, and all the faces are rectangular-shaped. Therefore, both the bases of a rectangular prism must also be rectangles. For example, we can relate the surface area of a rectangular prism with the surfaces of the objects given below: a book, cuboid-shaped aquarium, box, etc.

For example, let’s say there is a box that needs to be gift-wrapped. You need to calculate the amount of wrapping paper that will be needed to cover it. First, you will need to know the sum of the areas of the six surfaces of the box (four lateral side surfaces, top surface, and bottom surface). This total area of the six surfaces is called the surface area of a rectangular prism.

A rectangular prism can be classified as a right rectangular prism or an oblique rectangular prism. In the right rectangular prism, bases are perpendicular to each other whereas, in an oblique rectangular prism, the bases are not aligned one above the other. Let us see the basic formulas to calculate the total surface area and lateral surface area of a rectangular prism in the next section.

To calculate the surface area of a rectangular prism, we need to simply find the sum of the areas of the faces of the prism. A rectangular prism can have two types of areas:

• Total Surface Area
• Lateral Surface Area

Total Surface Area of Rectangular Prism

The total surface area of a rectangular prism can be calculated by finding the total area of all six faces. The formula to find the surface area of a rectangular prism is given as,

Total surface area of a rectangular prism = 2(lb + bh + lh) square units
where,

• l = Length of the rectangular prism
• b = Breadth of the rectangular prism
• h = Height of the rectangular prism

Note: The unit of measurement of the area of the rectangular prism is square units.

Lateral Surface Area of a Rectangular Prism

The lateral surface area of a rectangular prism can be calculated by finding the sum of all the lateral faces of the prism, i.e. the total area excluding the area of the bases. The formula to find the surface area of a rectangular prism is given as,

Lateral surface area of a rectangular prism = 2(l + b) h square units
where,

• l = Length
• b = Breadth
• h = Height of the rectangular prism

Example: Determine the total surface area of a rectangular prism with the following dimensions:

• Length (l) = 8 inches
• Breadth (b) = 5 inches
• Height (h) = 7 inches

Solution: The total surface area of the rectangular prism is given by: 2(lb + bh + lh) = 2(8 × 5 + 5 × 7 + 8 × 7) = 2(40 + 35 + 56) = 2 × 131 = 262 inch².

The surface area of a rectangular prism can be calculated using the following steps:

• Step 1: Check if the given dimensions of the rectangular prism are in the same units or not. If not, convert them into the same units.
• Step 2: Once the dimensions are in the same units, understand the need to calculate the lateral surface area or the total surface area according to the given situation.
• Step 3: Implement the formula for lateral surface area 2h(l + b), or total surface area, 2 (lb + bh + lh).
• Step 4: Write the unit as square units along with the values thus obtained.

Now that we know the formula and method to calculate the surface area of a rectangular prism, let us now understand how to calculate it with the help of an example.

Example: Determine the minimum area of the wrapping paper required to wrap a rectangular gift box with dimensions as given below:

• Length, l = 24 inches
• Breadth, b = 15 inches
• Height, h = 20 inches

Solution: In this case, we have to find the total surface area of the box. It can be calculated by following the steps given below:

Step 1: Determine the area of the top and bottom faces of the box.

The area of the bottom and the top surfaces are equal, therefore we can simply twice the area of the top or bottom of the box.

2lb = 2 × (24 × 15) = 2 × 360 = 720 inch²

Step 2: Next, we determine the area of the side faces of the box.

2bh = 2 × (15 × 20) = 2 × 300 = 600 inch²

2lh = 2 × (24 × 20) = 2 × 480 = 960 inch²

Step 3: Add the areas of all six faces to determine the total surface area of the box.

2(lb + bh + lh) = 2lb + 2bh + 2lh = 720 + 600 + 960 = 2280 inch².

Challenging Question:

Two boxes each of dimensions (4 inches × 5 inches × 6 inches) are joined together face to face to make a rectangular prism box. Determine the surface area of the newly formed box.

Tips and Tricks:

• A rectangular prism with all sides equal is called a cube.
• A rectangular prism in which the faces are not perpendicular to each other is called an oblique rectangular prism.
• If all the edges of a rectangular prism are equal, then the surface area of the prism will be equal to 6 times the area of each face.
• If all the edges of a rectangular prism are equal to ‘l’, then the volume of the prism is ‘l³‘.

Surface Area of Rectangular Prism Examples

Solved Examples

Question 1: Find the surface area of a rectangular prism with base 6 cm, h = 12 cm and side 5 cm.

Solution:

Given,
b = 6 cm
l = 5 cm
h = 12 cm

Surface area of a rectangular prism

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?

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.

Solved Examples on the Surface Area of a Rectangular Prism Formula

Example 1: Find the surface area of a rectangular prism if its length, breadth, and height are 4 m, 5 m, and 8 m.

Solution:

Given: l = 4 m, b = 5 m, and h = 8 m

SA = 2(lb + lh + bh)

= 2(4 × 5 + 4 × 8 + 5 × 8)

= 2(20 + 32 + 40)

= 2(92)

A = 184 m²

Example 2: Find the surface area of a rectangular prism if its length, breadth, and height are 2 m, 7 m, and 10 m.

Solution:

Given: l = 2 m, b = 7 m, and h = 10 m

SA = 2(lb + lh + bh)

= 2(2 × 7 + 2 × 10 + 7 × 10)

= 2(14 + 20 + 70)

= 2(104)

A = 208 m²

Example 3: Find the surface area of a rectangular prism if its length, breadth, and height are 9 m, 6 m, and 7 m.

Solution:

Given: l = 9 m, b = 6 m, and h = 7 m

SA = 2(lb + lh + bh)

= 2(9 × 6 + 7 × 9 + 6 × 7)

= 2(54 + 63 + 42)

= 2(159)

A = 318 m²

Example 4: Find the lateral surface area of a rectangular prism if its length, breadth, and height are 21 cm, 15 cm, and 18 cm.

Solution:

Given: l = 21 cm, b = 15 cm, and h = 18 cm

LSA = 2h (l+ b)

= 2 × 18 (21 +15)

= 36 × 36

LSA = 1,296 cm².

Example 5: Find the surface area of a rectangular prism if its length, breadth, and height are 7 m, 6 m, and 5 m.

Solution:

Given: l = 7 m, b = 6 m, and h = 5 m

SA = 2(lb + lh + bh)

= 2(7 × 6 + 7 × 5 + 6 × 5)

= 2(42 + 35 + 30)

= 2(107)

A = 214 m²

Example 6: Find the lateral surface area of a rectangular prism if its length, breadth, and height are 15 cm, 10 cm, and 13 cm.

Solution:

Given: l = 15 cm, b = 10 cm, and h = 13 cm

LSA = 2h (l+ b)

= 2 × 13 (15 + 10)

= 26 × 25

LSA = 650 cm².

Example 7: Find the surface area of a rectangular prism if its length, breadth, and height are 8 m, 5 m, and 9 m.

Solution:

Given: l = 8 m, b = 5 m, and h = 9 m

SA = 2(lb + lh + bh)

= 2(8 × 5 + 8 × 9 + 5 × 9)

= 2(40 + 72 + 45)

= 2(157)

A = 314 m²

Example 1

Earlier, you were given a problem about the candy factory workers, who need to figure out how much wrapper is needed to cover their famous candy bar.

First, plug the dimensions of the candy bar into the formula for surface area of a rectangular prism and multiply the values within the brackets:

Next, add the values together that are inside the parentheses;

The answer is the candy bar will require 34 square inches to completely cover the candy bar.

Example 2

What is the surface area of the figure below?

All of the faces of this prism are rectangles, so you can use the formula for finding the surface area of a rectangular prism as follows.

First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses:

Then, add the values together for the final answer. Remember to include the unit of measurement:

The answer is the rectangular prism has a surface area of 938 square centimeters. 

Example 3

Find the surface area of a rectangular prism with a length of 8 in, width of 4 inches, height of 6 inches.

First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses:

Then, add the values together for the final answer. Remember to include the unit of measurement:

The answer is the surface area for this rectangular prism is 208 square inches.

Example 4

Find the surface area of a rectangular prism with a length of 5 ft, width of 4 ft, height of 2 ft

First, plug the values given above into the surface area formula and multiply the values together within each of the parentheses: 

Then, add the values together for the final answer and include the unit of measurement:

The answer is the surface area for this rectangular prism is 76 square inches.

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