Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is derived from the fact that it is made by considering a polygonal base and expanding its sides as far as it creates an equilibrium with the opposing base.
This article post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also give examples of how to utilize the details provided.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The other faces are rectangles, and their number 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 both have an edge in parallel 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 broken down into these four parts:
A lateral face (implying both height AND depth)
Two parallel planes which constitute of each base
An illusory line standing upright across any provided 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 join
Types of Prisms
There are three main 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 sides. It looks 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 measure of the total amount of space that an object occupies. As an crucial 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, assuming,
V = Volume
B = Base area
h= Height
Ultimately, considering bases can have all kinds of figures, you will need to retain few formulas to determine the surface area of the base. However, we will touch upon 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 3D object 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 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 refers to 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 kind of prism.
Examples of How to Utilize the Formula
Considering we have the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s put them to use.
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 work on one more question, 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
As long as you have the surface area and height, you will figure 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 object is the measurement of the total area that the object’s surface comprises of. It is an essential part of the formula; therefore, we must understand how to find it.
There are a few different methods to figure out the surface area of a prism. To figure out 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 figure out the surface area of a triangular prism, we will use 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 Computing the Surface Area of a Rectangular Prism
Initially, we will determine 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 replace 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 find the surface area of a triangular prism, we will find the total surface area by following similar 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 information, you will be able to figure out any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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