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# Surface Area of a Prism(Definition, Formulas & Examples)

September 23, 2022

The surface area of a three-dimensional object is the total area of all its faces (including flat and curved). The surface area of a prism is the total area occupied by the faces of the prism and it depends upon the shape of its base. As the surface area denotes the area of the surface it is measured in terms of square units such as $m^{2}$, $cm^{2}$, $in^{2}$, $ft^{2}$.

Let’s learn how to find the surface area of a prism and its methods and formulas.

## Prism – A 3D Solid Shape

A prism is an important member of the polyhedron family that has congruent polygons at the base and top. The other faces of a prism are parallelograms are called lateral faces. It means that a prism does not have a curved face. A prism has the same cross-section all along its length. The prisms are named depending upon their cross-sections. The most common example of a prism is a metallic nut.

### Types of Prisms

The prisms are classified depending on the following factors

• Prisms based on the type of polygon, of the base
• Prisms based on the alignment of the identical bases
• Prisms based on the shape of the bases

A prism based on the type of the polygon of the base can be of the following two types.

• Regular Prism: If the base of the prism is in the shape of a regular polygon, the prism is a regular prism.
• Irregular Prism: If the base of the prism is in the shape of an irregular polygon, the prism is an irregular prism.

A prism based on the alignment of the identical bases can be of the following two types

• Right Prism: A right prism has two flat ends that are perfectly aligned with all the side faces in the shape of rectangles.
• Oblique Prism: An oblique prism appears to be tilted the two flat ends are not aligned and the side faces are parallelograms.

A prism is named on the basis of the shape obtained by the cross-section of the prism and can be any of the following types

• Triangular Prism: A prism whose bases are triangles in shape is considered a triangular prism.
• Square Prism: A prism whose bases are square in shape is considered a square prism. (a rectangular prism is cubical in shape)
• Rectangular Prism: A prism whose bases are rectangular in shape is considered a rectangular prism (a rectangular prism is cuboidal in shape).
• Trapezoidal Prism: A prism whose bases are trapezoid in shape is considered a trapezoidal prism.
• Pentagonal Prism: A prism whose bases are pentagon in shape is considered a pentagonal prism.
• Hexagonal Prism: A prism whose bases are hexagon in shape is considered a hexagonal prism.
• Octagonal Prism: A prism whose bases are octagon in shape is considered an octagonal prism.
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## What is the Surface Area of a Prism?

The area occupied by the surface/boundary of a prism is known as the surface area of a prism. It is always measured in square units such ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.).

Since a prism has two bases and $n$ number of parallelogram sides depending on the cross-section of the prism, there are two types of surface areas of a prism

• Lateral Surface Area
• Total Surface Area

### Lateral Surface Area of a Prism

The lateral surface area of a prism is the area of its vertical faces(or lateral faces, which are parallelograms), in case a prism has its bases facing up and down. Thus, the lateral surface area of a prism = $\left( \text{Base Perimeter} \right) \times \left( \text{Height} \right)$.

For a $n$-sided polygonal prism, the perimeter of the base is $n \times s$, where $n$ is the number of edges(or sides) of a polygonal base and $s$ is the length of each edge(or side).

Therefore, the lateral surface area of a prism is given by the formula $n \times s \times h$,

where $n$ is the number of edges(or sides) of a polygonal top or base

$s$ is the length of each edge(or side) of a polygonal top or base

$h$ is the height of the prism

### Total Surface Area of a Prism

The total surface area of a prism is the sum of all the faces of a prism. There are two types of faces in a prism.

• Base and top faces (shape is $n$-sided polygon)
• Lateral faces (shape is parallelogram)

$\text{The total surface area of a Prism} = \left( \text{Lateral surface area of a prism} \right) + \left( \text{Area of the two bases} \right)$

$= \left( \text{Lateral surface area} \right) + \left(2 \times \text{Base Area} \right) \text{ or} \left(2 \times \text{Base Area} \right) + \left( \text{Base perimeter} \times \text{height} \right)$.

Area of an $n$-sided polygon is given by the formula $\text{Area} = \frac {ns^{2}}{4 \tan \frac {180}{n}}$, therefore, area of the two bases of a prism is calculated as $2 \times \frac {ns^{2}}{4 \tan \frac {180}{n}}$

where $n$ is the number of edges(or sides) of a polygonal top or base

$s$ is the length of each edge(or side) of a polygonal top or base

Therefore, the formula for the total surface area of a prism is  $2 \times \frac {ns^{2}}{4 \tan \frac {180}{n}} + n \times s \times h = ns \left(\frac{2s}{4 \tan \frac {180}{n}} + h\right)$.

where $n$ is the number of edges(or sides) of a polygonal top or base

$s$ is the length of each edge(or side) of a polygonal top or base

$h$ is the height of the prism

### Examples

Ex 1: Find the lateral surface area of the prism.

The given prism is a triangular prism. The sides of the triangular base are $5 m$, $13 m$, and $12 m$.

The perimeter of the base = $5 + 13 + 12 = 30 m$.

Height of prism $h = 3 m$

Therefore, lateral surface area of the prism = $30 \times 3 = 90 m^{2}$.

Ex 2: Find the total surface area of the prism.

The given prism is a triangular prism. The sides of the triangular base are $3 cm$, $4 cm$, and $5 cm$.

The perimeter of the base = $3 + 4 + 5 = 12 cm$.

Height of prism $h = 4 cm$.

The lateral surface area of the prism = $12 \times 4 = 48 cm^{2}$.

Area of the triangular base = $\frac {1}{2} \times 3 \times 4 = 6 cm^{2}$

Sum of area of the two bases = $2 \times 6 = 12 cm^{2}$

Therefore, total surface area of the prism = $48 + 12 = 60 cm^{2}$.

Ex 3: Find the total surface area of the prism.

The base of the given base is a parallelogram.

The length of the parallelogram $l = 4 m$ and the width of the parallelogram $w = 2.5 m$.

Perimeter of the parallelogram base = $2 \left(l + w \right) = 2 \times \left( 4 + 2.5\right) = 2 \times 6.5 = 13 m$.

Height of the prism $h = 2 m$.

Lateral surface area = $13 \times 2 = 26 m^{2}$.

Area of parallelogram base = $\text{side} \times \text{height} = 4 \times 2 = 8 m^{2}$.

Sum of the area of two bases = $2 \times 8 = 16 m^{2}$.

Therefore, total surface area = $26 + 16 = 42 m^{2}$.

## Conclusion

The surface area of a 3D shape (solid object) is a measure of the total area of the object’s surface. There are two types of surface areas in a prism, viz., lateral surface area calculated using the formula $n \times s \times h$ and total surface area calculated using the formula $ns \left(\frac{2s}{4 \tan \frac {180}{n}} + h\right)$, where $n$, $s$ and $h$ are the number of edges(sides), length of each edge(or side) and height of a prism respectively.

## Practice Problems

1. Find the lateral surface area of the following prisms.

2. Find the total surface area of the prisms in the above figure.

## FAQs

### What is the surface area of a prism?

The amount of area occupied by a prism is referred to as the surface area of a prism. The surface area of the prism depends on the base area of the prism and the lateral surface area of the prism. The unit of the surface area of the prism is expressed in $m^{2}$, $cm^{2}$, $in^{2}$, or $ft^{2}$.

### What is the formula for the surface area of a prism?

There are two types of surface area in a prism.
a) Lateral Surface Area: The formula for lateral surface area is $\text{Perimeter of Base} \times \text{Height}$.
b) Total Surface Area: The formula for total surface area is $\left(\text{Perimeter of Base} \right) \times \text{Height} + 2 \times \left(\text{Area of the Base} \right)$

### How does the surface area of a prism change if the type of prism changes?

The surface area of the prism depends on the base area of the prism and the lateral surface area of the prism. Different types of prisms have different bases hence, as the type of prism changes, the base of the prism changes. This changes the base area of the prism changes which in turn changes the surface area of the prism.