Volume of a Prism

Volume of Prism Formula

We know that geometry deals with the study of 2D shapes and 3D shapes. A polyhedron is a three-dimensional shape that is formed by polygons. The word polyhedron is made of ‘poly’ meaning many and ‘hedron’ meaning face.


Every polygon in a polyhedron is called a face. The two faces meet in a line segment that is called edge. The point of intersection of three or more edges is called a vertex.

There are two important members of the polyhedron family: Prisms and pyramids. In this article, we will be discussing what is prisms, different types of prisms, the volume of prism formula, and how to calculate the volume of a prism.


What is Prism?

A three-dimensional solid shape having its base and top as identical polygons and side faces as parallelograms are called a prism. 


A Prism is a Solid Object with:

  • Identical base and top which are parallel to each other.

  • The side faces are flat and parallelogram

  • No curve sides

  • And the same cross-section along with its length.

 

Different Types of Prism

A prism is a solid three-dimensional geometric figure with two similar ends and all flat sides. The prism is named after the shape of its base, hence a prism with a triangular base is called a "triangular prism”. So the different types of prisms are given their names on the basis of their cross-sectional figure formed. 


Types of Prisms are:

  • Triangular Prism

  • Square Prism

  • Cube

  • Cuboid or rectangular Prism

  • Pentagonal Prism


What is Volume of a Prism?

As the prism is a 3D solid object it has both the surface area and volume.

The volume of a 3D prism is defined as the total space occupied by that object. 

To calculate the volume of a prism, you just have to calculate the area of its base and multiply it by its height.

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Therefore Volume General Formula it is Represented as,

Volume of a Prism (V) = Base Area × Length


The volume of a three-dimensional prism is represented as cubic units. 

Here's how to calculate the volume of a variety of prisms. 


Volume of Prism Formula

Different prisms have different volumes. So the formula to calculate the volume of different Prisms are:

  • Triangular Prism

A prism having its base and top as identical triangles and the lateral faces are rectangles is called a triangular prism.

A triangular prism has

  • 5 faces

  • 6 vertices and

  • 9 edges

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Volume of Triangular Prism = (\[\frac{1}{2}\]) a x b x h 


Where,

a = Apothem length of a triangular prism

b = Base length of a triangular prism

h = height of a triangular prism

  • Square Prism

In a square prism, the base and top are congruent squares and the lateral faces are rectangles


A square prism has

  • 6 faces

  • 8 vertices and

  • 12 edges

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Volume of a Square Prism = l x b x h


Where

l = length of a square prism

b = Base of a square prism

h = height of a square prism

  • Cube

If a square prism has all of its faces as identical squares, then it is called a cube prism.

A cubic prism or a cube has


  • 6 faces

  • 8 vertices and

  • 12 edges

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Volume of a Cube Prism = \[a^{3}\]


Where

a = edges of a cube prism( because l = w = h = a)

  • Cuboid or Rectangular Prism

If the base and top of the prism are identical rectangles, then it is a rectangular prism or a cuboid.


A cuboid prism has

  • 6 faces

  • 8 vertices and

  • 12 edges

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Volume of Rectangular Prism = l x b x h


Where

l = length of a rectangular prism

b = Base of a rectangular prism

h = height of a rectangular prism

  • Pentagonal Prism

If the base and top of a prism are pentagons, then it is called a pentagonal prism. 

A pentagonal prism has


  • 7 faces

  • 10 vertices and

  • 15 edges

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Volume of Pentagonal Prism = (\[\frac{5}{2}\]) a x b x h


Where,

a – Apothem length of the pentagonal prism.

b – Base length of the pentagonal prism.

h – Height of the pentagonal prism

  • Hexagonal Prism

A hexagonal prism is a prism with six rectangular faces and top and base as hexagonal.

A hexagonal prism has


  • 8 faces

  • 12 vertices and

  • 18 edges

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Volume of Hexagonal Prism = 3 x a x b x h


Where

a – Apothem length of the hexagonal prism.

b – Base length of the hexagonal prism.

h – Height of the hexagonal prism.


Solved Examples

Example 1 : Find the volume of the triangular prism given below.

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Solution: 

Given that a = Apothem length of a triangular prism = 9cm

b = Base length of a triangular prism = 12 cm

h = height of a triangular prism = 18cm

We have, Volume of triangular prism = (\[\frac{1}{2}\]) a x b x h 

=  \[\frac{1}{2}\] x 9 x 12 x 18 

= 972 \[cm^{3}\]

So the volume of the triangular prism is 972 cubic centimeter.


Example 2: Find the volume of rectangular prism given below.

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Solution:

Given that: l = length of a rectangular prism = 9cm

b = Base of a rectangular prism = 7cm

h = height of a rectangular prism = 13cm

We have, Volume of Rectangular Prism = l x b x h

                      = 9 x 7 x 13

          = 819 \[cm^{3}\]

Therefore the volume of rectangular prism is 819 cubic centimeters.


Quiz Time:

  1. Find the volume for the triangular prism given below

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FAQ (Frequently Asked Questions)

1. What is a Polygon?

Answer: A two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.


The below figures show some of the examples of polygons.

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In figure 1 you can see that all the shapes are polygons, as all the shapes are drawn joining the straight lines only. There are no curved lines. Even the shape in orange color is a polygon because it is formed by joining straight lines and it is a closed figure.

But in figure 2 the shape is not a polygon because it is not fully connected and also has curved lines.


The name of the polygon itself implies the number of sides in it. For instance, a triangle is a polygon having three sides, a quadrilateral is a polygon having four sides, etc.

2. What is Pyramid?

Answer: A solid shape having its base as any polygon and side face as triangles with a common vertex is called a pyramid. 


The most commonly used pyramid is the square pyramid i.e., it has a square base and four triangular side faces.


Some of the properties of a square pyramid are :

A square pyramid has 5 vertices, 8 edges, and 5 faces 

(image will be uploaded soon)Volume of Prism Formula