# Volume of a Prism - Formula, Derivation, Definition, Examples

A prism is a crucial figure in geometry. The shape’s name is originated from the fact that it is created by taking a polygonal base and extending its sides as far as it cross the opposing base.

This article post will take you through what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also give instances of how to employ the information provided.

## What Is a Prism?

A prism is a 3D geometric shape with two congruent and parallel faces, called bases, which take the form of a plane figure. The additional faces are rectangles, and their amount relies on how many sides the identical base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there will be five sides.

### Definition

The properties of a prism are interesting. The base and top both have an edge in common with the additional two sides, creating them congruent to each other as well! This implies that every three dimensions - length and width in front and depth to the back - can be broken down into these four parts:

A lateral face (signifying both height AND depth)

Two parallel planes which constitute of each base

An illusory line standing upright through any given point on either side of this shape's core/midline—also known collectively as an axis of symmetry

Two vertices (the plural of vertex) where any three planes join

### Kinds of Prisms

There are three primary types of prisms:

Rectangular prism

Triangular prism

Pentagonal prism

The rectangular prism is a common type of prism. It has six faces that are all rectangles. It looks like a box.

The triangular prism has two triangular bases and three rectangular faces.

The pentagonal prism has two pentagonal bases and five rectangular sides. It seems almost like a triangular prism, but the pentagonal shape of the base makes it apart.

## The Formula for the Volume of a Prism

Volume is a calculation of the total amount of area that an item occupies. As an essential shape in geometry, the volume of a prism is very relevant in your learning.

The formula for the volume of a rectangular prism is V=B*h, assuming,

V = Volume

B = Base area

h= Height

Consequently, since bases can have all types of figures, you have to retain few formulas to determine the surface area of the base. However, we will go through that afterwards.

### The Derivation of the Formula

To derive the formula for the volume of a rectangular prism, we are required to observe a cube. A cube is a 3D object with six faces that are all squares. The formula for the volume of a cube is V=s^3, assuming,

V = Volume

s = Side length

Now, we will have a slice out of our cube that is h units thick. This slice will make a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula refers to the base area of the rectangle. The h in the formula stands for height, that is how dense our slice was.

Now that we have a formula for the volume of a rectangular prism, we can use it on any type of prism.

### Examples of How to Use the Formula

Considering we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, now let’s use them.

First, let’s figure out the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.

V=B*h

V=36*12

V=432 square inches

Now, consider another question, let’s calculate the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.

V=Bh

V=30*15

V=450 cubic inches

Provided that you possess the surface area and height, you will figure out the volume without any issue.

## The Surface Area of a Prism

Now, let’s discuss about the surface area. The surface area of an item is the measure of the total area that the object’s surface consist of. It is an crucial part of the formula; thus, we must learn how to calculate it.

There are a several distinctive methods to work out the surface area of a prism. To calculate the surface area of a rectangular prism, you can use this: A=2(lb + bh + lh), where,

l = Length of the rectangular prism

b = Breadth of the rectangular prism

h = Height of the rectangular prism

To compute the surface area of a triangular prism, we will utilize this formula:

SA=(S1+S2+S3)L+bh

where,

b = The bottom edge of the base triangle,

h = height of said triangle,

l = length of the prism

S1, S2, and S3 = The three sides of the base triangle

bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh

We can also utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)

### Example for Finding the Surface Area of a Rectangular Prism

First, we will work on the total surface area of a rectangular prism with the ensuing information.

l=8 in

b=5 in

h=7 in

To solve this, we will plug these numbers into the corresponding formula as follows:

SA = 2(lb + bh + lh)

SA = 2(8*5 + 5*7 + 8*7)

SA = 2(40 + 35 + 56)

SA = 2 × 131

SA = 262 square inches

### Example for Computing the Surface Area of a Triangular Prism

To find the surface area of a triangular prism, we will find the total surface area by following same steps as priorly used.

This prism will have a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Therefore,

SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)

Or,

SA = (40*7) + (2*60)

SA = 400 square inches

With this knowledge, you will be able to calculate any prism’s volume and surface area. Check out for yourself and see how simple it is!

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