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Volume of a Prism(Formula, Derivation & Examples)

volume of a prism

A prism is an important member of the polyhedron family that has congruent polygons at the base and top. The space occupied by a prism is called the volume of a prism. It’s also referred to as the capacity of a prism and it depends on its base area and height. Volume of a prism is measured in terms of cubic units, such as $cm^{3}$, $m^{3}$, $mm^{3}$, $in^{3}$, $ft^{3}$, etc.

Let’s learn how to find the volume of a prism and its formula.

What is the Volume of a Prism?

The volume of a prism is the amount of space a prism occupies in a three-dimensional space. A prism is a solid 3-D shape that has two same polygonal faces and other lateral faces that are parallelograms. A prism is a polyhedron and different types of prisms are named according to the shape of their bases such as

  • triangular prism (triangular base) 
  • square prism (square base)
  • rectangular prism (rectangular base) 
  • pentagonal prism (pentagonal base)
  • hexagonal prism (hexagonal base)
  • octagonal prism (octagonal base)

In general, the volume of a prism is calculated using the formula $\text{Area of the Base} \times \text{Height of Prism}$.

The general formula for finding the area of a $n$-sided polygon is given by $\text{Area} = \frac {ns^{2}}{4 \tan \frac {180}{n}}$

where $n$ is the number of edges (or sides) in a polygon

$s$ is the length of the edge(or side) of a polygon

Therefore, the formula for volume of a $n$-sided prism of edge length $s$ and height $h$ is given by $\text{Volume} = \frac {ns^{2}h}{4 \tan \frac {180}{n}}$.

Volume Formula for Prisms Of Different Types

Prism TypeNumber of sides in Base $n$Volume Formula
Triangular Prism$3$$\frac {3s^{2}h}{4 \tan 60^{\circ}}$
Square Prism$4$$\frac {s^{2}h}{\tan 45^{\circ}}$
Pentagonal Prism$5$$\frac {5s^{2}h}{4 \tan 36^{\circ}}$
Hexagonal Prism$6$$\frac {3s^{2}h}{2 \tan 30^{\circ}}$
Heptagonal Prism$7$$\frac {7s^{2}h}{4 \tan 25.71^{\circ}}$
Octagonal Prism$8$$\frac {2s^{2}h}{\tan 22.5^{\circ}}$
Nonagonal Prism$9$$\frac {9s^{2}h}{4 \tan 20^{\circ}}$
Decagonal Prism$10$$\frac {5s^{2}h}{2 \tan 18^{\circ}}$

Note: 

  • The above formulae are applicable only when bases are regular polygons.
  • In case the bases are irregular polygons, then first find the area of the base and multiply it by the height of the prism to find the volume.

Examples

Ex 1: Find the volume of the following triangular prism.

volume of a prism

The base of the prism is a right triangle with each side $6 m$.

Area of triangular base $\text{B} = \frac {1}{2} \times 6 \times 6 = 18 m^{2}$.

Height of the prism $\text{H} = 9 m$.

The volume of the prism = $\text{B} \times \text{H} = 18 \times 9 = 162 m^{3}$.

Ex 2: Find the volume of the following triangular prism.

volume of a prism

The base of a prism is an equilateral triangle of sides $6 m$.

Area of the base of a prism $\text{B} = \frac {\sqrt{3}}{4} \times 6^{2} = \frac {\sqrt{3}}{4} \times 36 = 9 \sqrt{3} m^{2}$.

Height of a prism $\text{H} = 12 m$.

The volume of prism = $\text{B} \times \text{H} = 9 \sqrt{3} \times 12 = 108 \sqrt{3} m^{3}$.

Ex 3: Find the volume of the given prism.

volume of a prism

The base of a prism is a rectangle of length $6$ units and width $4$ units.

Area of the base of a prism = $\text{B} = lw = 6 \times 4 = 24 unit^{2}$.

Height of a prism $\text{H} = 2$ units.

The volume of prism = $\text{B} \times \text{H} = 24 \times 2 = 48 unit^{3}$.

Types of Coordinate Systems

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.

volume of a prism
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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.
volume of a 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.
volume of a prism

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 triangle 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.
volume of a prism

FAQs

What is the volume of a prism?

The amount of space occupied by a prism is referred to as the volume of a prism. The volume of the prism depends on the base area of the prism and the height of the prism. The unit of volume of the prism is expressed in $m^{3}$, $cm^{3}$, $in^{3}$, $ft^{3}, etc.

What is the formula for the volume of a prism?

The formula for the volume of a prism is obtained by finding the product of the base area and height of the prism. The volume of a prism is given as $\text{V} = \text{B} \times \text{H}$ where, $\text{V}$ is the volume of the prism, $\text{B}$ is the area of the base of the prism, and $\text{H}$ is the height of the prism.

How does the volume of the prism change if the type of prism changes?

The volume of the prism depends on the base area of the prism. As the type of prism changes, the base of the prism changes thereby changing the base area of the prism. This change in the base area of the prism changes the volume of the prism.

Conclusion

The volume of a prism is the number of cubic units, occupied by the prism completely and is calculated by finding the product of its base area and the height. Since the formula for finding the area of different polygons are different, so the formulas for calculating the volume of different types of prisms are different.

Practice Problems

Find the volume of the following prisms.

volume of a prism

Recommended Reading

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