Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The figure’s name is derived from the fact that it is made by taking into account a polygonal base and expanding its sides until it intersects the opposite 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 offer instances of how to utilize the data provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, well-known as bases, that take the shape of a plane figure. The additional faces are rectangles, and their count 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 characteristics of a prism are fascinating. The base and top each have an edge in common with the other two sides, creating them congruent to one another as well! This states that all three dimensions - length and width in front and depth to the back - can be deconstructed 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 across any provided point on any side of this figure's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
Kinds of Prisms
There are three main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type of prism. It has six faces that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It looks almost like 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 total amount of area that an object occupies. As an important figure 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
Ultimately, since bases can have all types of shapes, you will need to know a few formulas to figure out 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 have 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
Immediately, we will take 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 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 kind of prism.
Examples of How to Utilize the Formula
Now that we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s utilize these now.
First, let’s calculate 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, consider 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
Considering that you have the surface area and height, you will work out the volume without any issue.
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 occupies. It is an crucial part of the formula; thus, we must learn how to calculate it.
There are a few distinctive ways to work out the surface area of a prism. To measure 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 calculate 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 utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Calculating 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 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 compute the surface area of a triangular prism, we will find the total surface area by following same steps as before.
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 will be able to compute any prism’s volume and surface area. Try it out for yourself and see how easy it is!
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