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 created by taking into account a polygonal base and expanding its sides until it cross the opposite base.
This blog post will talk about what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also provide examples of how to utilize the data provided.
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 additional faces are rectangles, and their number relies 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 fascinating. The base and top both 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 make up each base
An imaginary line standing upright across any given point on either 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 join
Kinds of Prisms
There are three main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type of prism. It has six faces 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 seems a lot 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 measure of the sum of area that an item occupies. As an essential shape 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
Ultimately, given that bases can have all types of shapes, you will need to learn few formulas to determine the surface area of the base. Despite that, we will touch upon that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we need to observe a cube. A cube is a 3D item 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
Now, we will take a slice out of our cube that is h units thick. This slice will by itself be 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 implies the height, that is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any type of prism.
Examples of How to Utilize the Formula
Now that we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, now let’s use them.
First, let’s figure out 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 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 figure out the volume with no problem.
The Surface Area of a Prism
Now, let’s discuss about 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; thus, we must understand how to find 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 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 figure out the surface area of a triangular prism, we will use 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 work on 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 put 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 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 same 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 will be able to calculate any prism’s volume and surface area. Check out for yourself and see how easy it is!
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