Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The figure’s name is originated from the fact that it is created by considering a polygonal base and stretching its sides as far as it creates an equilibrium with the opposite base.
This article post will take you through what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also give instances of how to use the details given.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, known as bases, which take the shape of a plane figure. The other faces are rectangles, and their count 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 characteristics of a prism are fascinating. The base and top each have an edge in common with the other two sides, creating them congruent to one another as well! This means 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 (meaning both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any provided point on either side of this shape's core/midline—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 kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common kind of prism. It has six faces that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It seems a lot like a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a measurement of the sum of area that an object occupies. As an essential 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, where,
V = Volume
B = Base area
h= Height
Consequently, considering bases can have all types of figures, you will need to retain few formulas to figure out the surface area of the base. Still, we will touch upon that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional object with six faces that are all squares. The formula for the volume of a cube is V=s^3, where,
V = Volume
s = Side length
Now, we will get 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 implies the base area of the rectangle. The h in the formula stands for height, that is how thick 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 Utilize the Formula
Now that we have the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, now let’s use them.
First, let’s work on 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, let’s try another problem, let’s work on 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 have the surface area and height, you will figure out the volume with no issue.
The Surface Area of a Prism
Now, let’s discuss regarding the surface area. The surface area of an object is the measure of the total area that the object’s surface occupies. It is an essential part of the formula; consequently, we must understand how to calculate it.
There are a few distinctive ways to figure out the surface area of a prism. To figure out the surface area of a rectangular prism, you can employ 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
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
First, we will determine the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To solve this, we will plug these values 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 figure out the total surface area by following similar 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 figure out any prism’s volume and surface area. Try it out for yourself and see how easy it is!
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