Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The figure’s name is originated from the fact that it is created by considering a polygonal base and expanding its sides as far as it cross the opposite base.
This blog post will take you through what a prism is, its definition, different types, and the formulas for volume and surface area. We will also give instances of how to employ the data given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, well-known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their number rests on how many sides the similar 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 properties of a prism are interesting. The base and top both have an edge in common with the other two sides, making them congruent to each other as well! This means 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 (meaning both height AND depth)
Two parallel planes which constitute of each base
An illusory line standing upright through 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 meet
Types of Prisms
There are three main types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common 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 sides.
The pentagonal prism has two pentagonal bases and five rectangular faces. 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 calculation of the sum of space that an item occupies. As an important 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
Finally, since bases can have all sorts of shapes, you are required to retain few formulas to figure out the surface area of the base. Despite that, we will touch upon that later.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we have to look at 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 by itself be 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 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
Now that we know the formulas for the volume of a pentagonal prism, triangular prism, and rectangular 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 try another 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
Considering that you have the surface area and height, you will work out the volume with no problem.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an item is the measurement of the total area that the object’s surface occupies. It is an essential part of the formula; therefore, we must learn how to find it.
There are a few distinctive ways 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 compute 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 Finding the Surface Area of a Rectangular Prism
First, we will figure out 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 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 Finding 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 identical 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 compute any prism’s volume and surface area. Test it out for yourself and observe how easy it is!
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