Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is derived from the fact that it is made by considering a polygonal base and expanding its sides till it intersects the opposing 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 offer examples of how to use the information given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, called bases, that take the shape of a plane figure. The additional faces are rectangles, and their number 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 properties of a prism are astonishing. The base and top both have an edge in parallel with the additional two sides, making them congruent to one another as well! This states that every three dimensions - length and width in front and depth to the back - can be decrypted into these four parts:
A lateral face (implying both height AND depth)
Two parallel planes which make up each base
An illusory line standing upright across any provided point on either side of this figure's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
Types of Prisms
There are three primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type 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 consists of two pentagonal bases and five rectangular faces. It looks 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 measurement of the total amount of area that an item occupies. As an crucial shape in geometry, the volume of a prism is very important for your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Finally, considering bases can have all types of figures, you will need to know a few formulas to determine the surface area of the base. Still, we will go through that later.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we need to observe 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, assuming,
V = Volume
s = Side length
Right away, we will take 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 dense 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 Utilize the Formula
Considering we know the formulas for the volume of a triangular prism, rectangular prism, and pentagonal 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, let’s try 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
As long as 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 regarding the surface area. The surface area of an item is the measure of the total area that the object’s surface occupies. It is an crucial part of the formula; thus, we must learn how to calculate it.
There are a few different ways 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 employ 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 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 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 replace 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 find the total surface area by ensuing same steps as before.
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 knowledge, you will be able to figure out any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
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