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# Lateral Surface Area of a Prism

There are many three-dimensional shapes that we come across in our daily life. Among them, the shapes whose faces are polygons, are known as polyhedrons. A prism is a kind of polyhedron in which there are two parallel and congruent bases. It is made up of polygonal lateral faces joining two bases. The cube and cuboid are the examples of prisms.

The prisms can be classified mainly into two types - right prisms and oblique prisms. In a right prism, the bases are located exactly one above another. In other words, if we draw a line that joins center of the both bases, then it would be perpendicular to each other. The side faces of a right prism are always rectangles. Right rectangular and right pentagonal prisms are shown below.

The oblique prisms typically do not have both bases one above another. The shape of an oblique prism looks tilted to one side. The side faces of such prisms are parallelograms. An oblique triangular prism is demonstrated in the following diagram.

We can clearly see that the perpendicular drawn from the center of one base does not meet in the center of another base.

In this article, we are going to focus only on the lateral surface area of prisms. We will discuss the formula and method of calculating it. Let us go ahead and learn this particular concept about prisms in detail.

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## Definition

The area of something is the amount of surface that it covers. In solid figures, we generally calculate two types of areas - lateral surface area and total surface area. The lateral surface area is the area of all the lateral faces or side faces of a three-dimensional object, while total surface area includes the area of lateral faces as well as the area of a base.

Similarly, for a prism, the lateral surface area will be the area of all the lateral surfaces. It eventually excludes the area of the two bases. It means that lateral surface area of a prism is the area of all non-base faces. For a rectangular prism, any two faces can serve as the bases. So, the lateral area will be the areas of any 4 rectangles which do not act as the bases at that moment. In triangular and pentagonal prisms, the areas of two triangles and two pentagons will not be included while calculating the lateral surface area.

## Formula

For a prism, the lateral surface area is calculated by finding the product of perimeter of the base by the height of the prism. The formula for lateral surface of prism can be framed as under :

Lateral surface area = Perimeter of base x height

This can be abbreviated as below :

LSA = P h

Recall that the perimeter of any 2D shape is the total length of its outer boundary, while the height of a prism is the length of perpendicular drawn from a point on one base to the other.

Note : This formula works only for right prisms. For finding the lateral surface area of an oblique prism, one has to calculate the area of each side face separately and then add them.

## Rules

Rules regarding finding the lateral surface area of a prism are:

1) The formula is applicable on right prisms only since oblique prisms may have different areas of each side face.

2) The height is the shortest distance between two bases in a right prism.

3)
The perimeter and height both must be represented by the same unit before calculating the lateral surface area.

4)
The unit of lateral surface area is the unit of area, i.e unit$^{2}$.

## How To Find Lateral Surface Area of a Prism

How do you execute above formula practically ?

Steps to find the lateral surface area
of a prism:

Step 1 : Determine the height of given prism. Generally, it would be given in the question itself or you have to interpret it through the diagram.

Step 2 : Calculate the perimeter of the base. For this, determine what kind of base it has. Apply the formula of perimeter according to the shape of the base. Otherwise, just add lengths of all the sides of the base.

Step 3 : Find the product of perimeter and height so obtained.

Step 4 : Write the answer with appropriate unit of area which is square unit or unit$^{2}$.

## Examples

Refer the following examples in order to have a better understanding.

Example 1 : Find the lateral surface area of a triangular prism whose height is 7 cm. Given that the base is an equilateral triangle of 5 cm side.

Solution : Determining Height

Given, h = 7 cm

Finding Perimeter

Perimeter of base = Perimeter equilateral triangle

P = 3 $\times$ side = 3 $\times$ 5 = 15 cm

Plug Values in Formula

LSA = P h

LSA = 15 $\times$ 7

Solve

LSA = 15 $\times$ 7 = 105 cm$^{2}$
Example 2 : Calculate the height of a rectangular prism whose base is a square of 3.5 mm side and the lateral surface area is 112 square mm.

Solution : Determine lateral surface area

Given that LSA = 112 mm$^{2}$

Calculate perimeter

P = perimeter of square

P = 4 $\times$ side = 4 $\times$ 3.5 = 14 mm

Plug Values in Formula

LSA = P h

112 = 14 $\times$ h

Manipulate and Calculate

h = $\frac{112}{14}$

h = 8 mm
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