Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The shape’s name is derived from the fact that it is created by considering a polygonal base and stretching its sides until it cross the opposite base.
This blog post will talk about what a prism is, its definition, different types, and the formulas for volume and surface area. We will also take you through some examples of how to use the data given.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, known as bases, that take the shape of a plane figure. The other faces are rectangles, and their count rests 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 will be five sides.
Definition
The characteristics of a prism are fascinating. The base and top each have an edge in common with the additional two sides, creating them congruent to each other as well! This implies that all three dimensions - length and width in front and depth to the back - can be decrypted into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any given point on either side of this shape'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 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 matches the looks of 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 a lot 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 measurement of the sum of space that an thing occupies. As an crucial figure 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, where,
V = Volume
B = Base area
h= Height
Finally, since bases can have all sorts of shapes, you have to know a few formulas to calculate 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 need 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, 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 stands for 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 use it on any type of prism.
Examples of How to Utilize the Formula
Since we understand the formulas for the volume of a triangular prism, rectangular prism, and pentagonal 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 one more 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 have the surface area and height, you will work out the volume with no 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 occupies. It is an important part of the formula; therefore, we must know how to find it.
There are a several varied ways to figure out the surface area of a prism. To calculate the surface area of a rectangular prism, you can utilize this: A=2(lb + bh + lh), assuming,
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 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 use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Calculating the Surface Area of a Rectangular Prism
First, we will work on the total surface area of a rectangular prism with the ensuing dimensions.
l=8 in
b=5 in
h=7 in
To solve this, we will replace these values 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 work on the total surface area by ensuing same steps as earlier.
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 data, you should be able to compute any prism’s volume and surface area. Test it out for yourself and observe how easy it is!
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