Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The shape’s name is originated from the fact that it is created by taking a polygonal base and extending its sides till it creates an equilibrium with the opposite base.
This blog post will take you through what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also give examples of how to utilize the details given.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, well-known as bases, which 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 will be five sides.
Definition
The characteristics of a prism are fascinating. The base and top each have an edge in parallel with the other two sides, making 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 make up each base
An illusory line standing upright through any given point on any side of this figure's core/midline—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Kinds of Prisms
There are three major types 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 looks like a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism consists of two pentagonal bases and five rectangular faces. It seems close to a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a calculation of the total amount of area that an object occupies. As an important 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 sorts of figures, you have to learn few formulas to figure out the surface area of the base. Still, we will go through that afterwards.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to observe a cube. A cube is a three-dimensional item 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
Now, we will have a slice out of our cube that is h units thick. This slice will create a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula stands for the base area of the rectangle. The h in the formula refers to 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 understand the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s put them to use.
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 one more problem, 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
As long as you possess the surface area and height, you will work 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 measurement of the total area that the object’s surface occupies. It is an important part of the formula; consequently, we must learn how to calculate it.
There are a few varied ways to find 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 compute the surface area of a triangular prism, we will employ 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 use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding the Surface Area of a Rectangular Prism
Initially, we will figure out the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To calculate this, we will put 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 Computing the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will work on the total surface area by following same steps as earlier.
This prism consists of 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 information, you will be able to compute any prism’s volume and surface area. Test it out for yourself and see how easy it is!
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