Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The figure’s name is derived from the fact that it is created by taking a polygonal base and stretching its sides until it cross the opposite base.
This article post will take you through what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also provide examples of how to utilize the details given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, known as bases, that take the form of a plane figure. The other faces are rectangles, and their number relies 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 will be five sides.
Definition
The properties of a prism are fascinating. The base and top both have an edge in common with the additional two sides, making them congruent to each other as well! This means 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 (meaning both height AND depth)
Two parallel planes which constitute of each base
An imaginary line standing upright across any provided point on any side of this shape's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Kinds of Prisms
There are three primary types 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 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 seems 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 object occupies. As an essential shape in geometry, the volume of a prism is very relevant in your studies.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all sorts of figures, you will need to know a few formulas to figure out the surface area of the base. Despite that, 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 look at 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, where,
V = Volume
s = Side length
Now, we will have 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 stands for the base area of the rectangle. The h in the formula implies the height, which is how dense 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 pentagonal prism, triangular prism, and rectangular 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, let’s try 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 figure out the volume without any issue.
The Surface Area of a Prism
Now, let’s discuss 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 essential part of the formula; therefore, we must know how to find it.
There are a few different ways 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 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
First, we will figure out the total surface area of a rectangular prism with the ensuing dimensions.
l=8 in
b=5 in
h=7 in
To calculate 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 Finding the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will figure out the total surface area by ensuing identical steps as priorly used.
This prism will have 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 data, you should be able to figure out any prism’s volume and surface area. Check out for yourself and see how easy it is!
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