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 considering a polygonal base and expanding its sides as far as it cross the opposite base.
This article post will discuss what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also provide examples of how to employ the details given.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, that take the form of a plane figure. The additional faces are rectangles, and their number 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 characteristics of a prism are interesting. The base and top each have an edge in parallel with the additional two sides, creating 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 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—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
Kinds of Prisms
There are three major 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 matches the looks of 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 looks 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 total amount of space that an object occupies. As an essential shape 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, where,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all types of figures, you will need to know a few formulas to determine the surface area of the base. However, we will touch upon that later.
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 item with six faces that are all squares. The formula for the volume of a cube is V=s^3, where,
V = Volume
s = Side length
Now, 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 refers to the base area of the rectangle. The h in the formula implies the height, that 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 Use the Formula
Considering we have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s utilize these now.
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 problem, 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
Considering that you possess the surface area and height, you will work out the volume with no problem.
The Surface Area of a Prism
Now, let’s discuss regarding 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; therefore, we must learn how to calculate 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), where,
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 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 Computing the Surface Area of a Rectangular Prism
Initially, we will work on the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To calculate this, we will plug 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 Finding the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will figure out the total surface area by ensuing similar steps as earlier.
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 knowledge, you will 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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