Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The shape’s name is originated from the fact that it is made by considering a polygonal base and expanding its sides until it cross the opposite base.
This article post will talk about what a prism is, its definition, different types, and the formulas for volume and surface area. We will also give examples of how to use the data 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 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 would 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 one another as well! This means that every 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 through any given point on any side of this shape's core/midline—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Types of Prisms
There are three major kinds 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 sides.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It appears 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 sum of space that an item occupies. As an essential figure in geometry, the volume of a prism is very important for your studies.
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 shapes, you have to learn few formulas to figure out the surface area of the base. However, we will go through that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we need to observe a cube. A cube is a three-dimensional 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 get a slice out of our cube that is h units thick. This slice will make 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, 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 type of prism.
Examples of How to Utilize the Formula
Considering we understand 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 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 calculate the volume with no issue.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an object is the measurement of the total area that the object’s surface occupies. It is an crucial part of the formula; consequently, we must learn how to find it.
There are a several different methods to work 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 calculate the surface area of a triangular prism, we will utilize 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 data.
l=8 in
b=5 in
h=7 in
To calculate this, we will put 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 Computing the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will work on the total surface area by ensuing same steps as earlier.
This prism will have 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 data, you should be able to calculate any prism’s volume and surface area. Check out for yourself and observe how simple it is!
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