Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is derived from the fact that it is made by taking a polygonal base and extending its sides till it creates an equilibrium with the opposing base.
This blog post will discuss what a prism is, its definition, different types, and the formulas for volume and surface area. We will also offer instances of how to employ the details given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, that take the form of a plane figure. The additional faces are rectangles, and their count depends 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 would be five sides.
Definition
The properties of a prism are interesting. The base and top each have an edge in common with the other two sides, creating them congruent to each other as well! This means that all three dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
A lateral face (implying both height AND depth)
Two parallel planes which constitute of each base
An illusory line standing upright through any provided point on either side of this shape's core/midline—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Types of Prisms
There are three main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular kind of prism. It has six sides that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular faces. It seems almost like a triangular prism, but the pentagonal shape of the base sets it apart.
The Formula for the Volume of a Prism
Volume is a calculation of the total amount of space that an item occupies. As an crucial figure 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, given that bases can have all types of figures, you will need to know a few formulas to figure out the surface area of the base. Despite that, we will go through that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we need to look at a cube. A cube is a 3D object with six sides that are all squares. The formula for the volume of a cube is V=s^3, assuming,
V = Volume
s = Side length
Immediately, we will get 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 implies the base area of the rectangle. The h in the formula refers to height, which is how dense our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any kind of prism.
Examples of How to Use the Formula
Considering we have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s utilize these now.
First, let’s calculate 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 figure out 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 calculate the volume without any issue.
The Surface Area of a Prism
Now, let’s discuss regarding the surface area. The surface area of an object is the measurement of the total area that the object’s surface consist of. It is an important part of the formula; therefore, we must understand how to calculate it.
There are a few varied methods to work out the surface area of a prism. To figure out the surface area of a rectangular prism, you can utilize 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 work out the surface area of a triangular prism, we will utilize this formula:
SA=(S1+S2+S3)L+bh
assuming,
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 Calculating the Surface Area of a Rectangular Prism
Initially, we will figure out the total surface area of a rectangular prism with the ensuing information.
l=8 in
b=5 in
h=7 in
To figure out this, we will plug these numbers into the respective 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 Calculating the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will find the total surface area by following similar 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. Hence,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
Or,
SA = (40*7) + (2*60)
SA = 400 square inches
With this data, you should be able to work out any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
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