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 taking into account a polygonal base and extending its sides as far as it cross the opposing base.
This blog post will discuss what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also take you through some examples of how to use the data provided.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, well-known as bases, that take the shape of a plane figure. The other faces are rectangles, and their count depends 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 would be five sides.
Definition
The characteristics of a prism are astonishing. The base and top both have an edge in common with the additional two sides, creating 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 broken down 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 meet
Kinds of Prisms
There are three major 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 matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. 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 total amount of area that an thing occupies. As an important figure in geometry, the volume of a prism is very relevant in your studies.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Ultimately, since bases can have all types of shapes, you have to retain few formulas to determine the surface area of the base. Despite that, we will touch upon 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 3D item with six faces that are all squares. The formula for the volume of a cube is V=s^3, where,
V = Volume
s = Side length
Now, we will have 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 refers to the base area of the rectangle. The h in the formula refers to height, that is how dense 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
Considering we have 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 work on one more 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 work 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 essential part of the formula; consequently, we must understand how to calculate it.
There are a few distinctive ways to find 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 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 utilize 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 determine 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 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 Computing the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will figure out the total surface area by following similar steps as priorly used.
This prism consists of a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Thus,
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 should be able to figure out any prism’s volume and surface area. Try it out for yourself and observe how easy it is!
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