Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The figure’s name is derived from the fact that it is made by taking into account a polygonal base and extending its sides until it creates an equilibrium with the opposing base.
This article post will take you through what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also provide 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, which take the shape of a plane figure. The other faces are rectangles, and their amount depends 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 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 states that all three dimensions - length and width in front and depth to the back - can be broken down into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright across any given point on either 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 join
Types of Prisms
There are three primary 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 has two pentagonal bases and five rectangular sides. It seems almost 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 sum of area that an thing occupies. As an important shape 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, given that bases can have all kinds of shapes, you have to learn few formulas to determine the surface area of the base. Despite that, we will go through that later.
The Derivation of the Formula
To extract 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 sides that are all squares. The formula for the volume of a cube is V=s^3, assuming,
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 stands for the base area of the rectangle. The h in the formula stands for height, which 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 kind of prism.
Examples of How to Utilize the Formula
Since we understand 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, consider 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 have 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 measurement of the total area that the object’s surface comprises of. It is an essential part of the formula; consequently, we must learn how to calculate it.
There are a several varied ways to work out the surface area of a prism. To measure 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 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 utilize 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 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 calculate the surface area of a triangular prism, we will find the total surface area by following identical steps as before.
This prism consists of 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 information, you should be able to calculate any prism’s volume and surface area. Check out for yourself and observe how easy it is!
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