Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The figure’s name is originated from the fact that it is created by taking into account a polygonal base and stretching its sides until it intersects the opposite base.
This article 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 employ the information provided.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, which 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 properties 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 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 (meaning both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright through any given point on either side of this shape's core/midline—also 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 type 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 looks almost like a triangular prism, but the pentagonal shape of the base makes it apart.
The Formula for the Volume of a Prism
Volume is a calculation of the total amount of space that an thing occupies. As an crucial figure in geometry, the volume of a prism is very important for your learning.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Consequently, given that bases can have all sorts of shapes, you have to learn few formulas to figure out the surface area of the base. Still, we will touch upon that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we need to look at 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, where,
V = Volume
s = Side length
Right away, we will have 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 use it on any type of prism.
Examples of How to Use the Formula
Since 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 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
Provided that you possess the surface area and height, you will calculate 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 measure of the total area that the object’s surface comprises of. It is an important part of the formula; thus, we must know how to find it.
There are a few different methods to find the surface area of a prism. To calculate the surface area of a rectangular prism, you can employ 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 work out the surface area of a triangular prism, we will employ 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
Initially, we will work on 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 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 Finding the Surface Area of a Triangular Prism
To calculate the surface area of a triangular prism, we will find the total surface area by ensuing same 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 knowledge, you will be able to calculate any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
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