# Volume of Prisms – Explanation & Examples

The volume of a prism is the total space occupied by a prism. In this article, you will learn how to find a prism volume by using the volume of a prism formula.

Before we get started, let’s first discuss what a prism is. By definition, a prism is a geometric solid figure with two identical ends, flat faces, and the same cross-section all along its length.

Prisms are named after the shapes of their cross-section. For example, a prism with a triangular cross-section is known as a triangular prism. Other examples of prisms include rectangular prism. pentagonal prism, hexagonal prism, trapezoidal prism etc.

## How to Find the Volume of a Prism?

To find the volume of a prism, you require the area and the height of a prism.  The volume of a prism is calculated by multiplying the base area and the height. The volume of a prism is also measured in cubic units, i.e., cubic meters, cubic centimeters, etc.

## Volume of a Prism Formula

The formula for calculating the volume of a prism depends on the cross-section or base of a prism. Since we already know the formula for calculating the area of polygons, finding the volume of a prism is as easy as pie.

The general formula for the volume of a prism is given as;

The volume of a Prism = Base Area × Length

Where Base is the shape of a polygon that is extruded to form a prism.

Let’s discuss the volume of different types of prisms.

### Volume of a triangular prism

A triangular prism is a prism whose cross-section is a triangle.

The formula for the volume of a triangular prism is given as;

Volume of a triangular prism = ½ abh

where,

a = apothem of a triangular prism.

The polygon’s apothem is the line connecting the polygon center to the midpoint of one of the polygon’s sides. The apothem of a triangle is the height of a triangle.

b = base length of a triangle

h = height of a prism.

Example 1

Find the volume of a triangular prism whose apothem is 12 cm, the base length is 16 cm and height, is 25 cm.

Solution

By the formula of a triangular prism,

volume = ½ abh

= ½ x 12 x 16 x 25

= 150 cm3

Example 2

Find the volume of a prism whose height is 10 cm, and the cross-section is an equilateral triangle of side length 12 cm.

Solution

Find the apothem of the triangular prism.

By Pythagorean theorem,

h2 + 62 =122

h2 + 36 =144

h2 = 108

h = 10.4 cm

Therefore, the apothem of the prism is 10.4 cm

Volume = ½ abh

= ½ x 10.4 x 12 x 10

= 624 cm3

### Volume of a pentagonal prism

For a pentagonal prism, the volume is given by the formula:

Volume of a pentagonal prism = (5/2) abh

Where,

a = apothem of a pentagon

b = base length of a pentagonal prism

h = height of a prism.

Example 3

Find the volume of a pentagonal prism whose apothem is 10 cm, the base length is 20 cm and height, is 16 cm.

Solution

Volume of a pentagonal prism = (5/2) abh

= (5/2) x 10 x 20 x 16

= 8000 cm3

### Volume of a hexagonal prism

A hexagonal prism has a hexagon as the base or cross-section. The volume of a hexagonal prism is given by:

Volume of a hexagonal prism = 3abh

where,

a = apothem length of a hexagon

b = base length of a hexagonal prism

h = height of a prism.

Example 4

Calculate the volume of a hexagonal prism with the apothem as 5 m, base length as 12 m, and height as 6 m.

Solution

Volume of a hexagonal prism = 3abh

= 3 x 5 x 12 x 6

= 1080 m3.

Alternatively, if the apothem of a prism is not known, then the volume of any prism is calculated as follows;

Volume of a prism = (h)(n) (s2)/ [4 tan (180/n)]

Where h = height of a prism

s = side length of the extruded regular polygon.

n = number of sides of a polygon

tan = tangent:

NOTE: This formula is only applied where the base or the cross-section of a prism is a regular polygon.

Example 5

Find the volume of a pentagonal prism with a height of 0.3 m and a side length of 0.1 m.

Solution

In this case, n = 5,

h = 0.3 m and   s = 0.1 m

By substitution,

Volume of a pentagonal prism = (0.3) (5) (0.12)/ [4 tan (180/5)]

= 0.015/4 tan 36

= 0.015/2.906

= 0.00516 m3.

### Practice Questions

1. Which of the following has a triangular base or cross-section?

2. Which of the following describes the cross-section of a hexagonal prism?

3. What is the volume of a triangular prism whose apothem is $16$ mm, the base length is $24$ mm and the height is $15$ mm?

4. A prism has a height of $40$ ft and a cross-section that is an equilateral triangle measuring $20$ ft on each side. What is the volume of the prism shown below?

5. What is the volume of a pentagonal prism with: an apothem of $12$ m, a base length of $20$ m, and a height of $16$ m?

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