Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The figure’s name is derived from the fact that it is made by considering a polygonal base and expanding its sides until it creates an equilibrium with the opposite base.
This blog 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 employ the data given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, which take the shape of a plane figure. The additional faces are rectangles, and their count rests on how many sides the identical base has. For example, 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 astonishing. The base and top both have an edge in common with the other two sides, creating them congruent to one another as well! This states that all three dimensions - length and width in front and depth to the back - can be deconstructed into these four parts:
A lateral face (signifying both height AND depth)
Two parallel planes which make up each base
An fictitious line standing upright through any given point on any side of this shape's core/midline—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 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 faces.
The pentagonal prism has 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 sum of area that an object 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, where,
V = Volume
B = Base area
h= Height
Finally, given that bases can have all types of shapes, you are required to retain few formulas to determine the surface area of the base. Still, we will go through that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we are required to look at 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, where,
V = Volume
s = Side length
Right away, 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 stands for height, which 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 kind of prism.
Examples of How to Use the Formula
Now that we have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, now let’s use them.
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, let’s work on another 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 calculate the volume with no problem.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an item is the measurement of the total area that the object’s surface comprises of. It is an essential part of the formula; consequently, we must learn how to calculate it.
There are a few different ways to figure 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), assuming,
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 figure out the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To figure out 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 Finding the Surface Area of a Triangular Prism
To calculate the surface area of a triangular prism, we will figure out the total surface area by ensuing 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. 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 information, you will be able to work out any prism’s volume and surface area. Test it out for yourself and see how easy it is!
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