Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is derived from the fact that it is made by taking into account a polygonal base and extending its sides as far as it intersects the opposite 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 give examples of how to use the details given.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their number depends on how many sides the similar 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 astonishing. The base and top each have an edge in common with the additional two sides, creating them congruent to one another as well! This means that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which constitute of each base
An illusory line standing upright through any given point on any 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 major kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular kind of prism. It has six faces that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism has two pentagonal bases and five rectangular faces. 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 measurement of the total amount of space that an object 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, assuming,
V = Volume
B = Base area
h= Height
Ultimately, since bases can have all kinds of figures, you will need to know a few formulas to determine the surface area of the base. Despite that, we will go through that later.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we have to observe a cube. A cube is a three-dimensional 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
Now, we will get a slice out of our cube that is h units thick. This slice will create 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 stands for height, which 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
Since 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, let’s try another question, let’s calculate 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 consist of. It is an crucial part of the formula; consequently, we must learn how to calculate it.
There are a few distinctive methods to work out the surface area of a prism. To calculate 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 use 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 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 determine the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To figure out this, we will plug 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 Computing 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 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. Check out for yourself and observe how simple it is!
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