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 made by taking a polygonal base and extending its sides till it cross the opposing base.
This blog post will talk about 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 given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their amount relies 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 properties of a prism are fascinating. The base and top both have an edge in parallel with the additional two sides, creating them congruent to one another as well! This states that every 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 make up each base
An illusory line standing upright across any given point on any side of this figure'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 primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common 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 sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular faces. It appears close to a triangular prism, but the pentagonal shape of the base stands out.
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 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, considering bases can have all kinds of shapes, you have to know a few formulas to calculate the surface area of the base. However, we will touch upon that afterwards.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we need to look at a cube. A cube is a 3D 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 take a slice out of our cube that is h units thick. This slice will make 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 type of prism.
Examples of How to Utilize the Formula
Now that we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s put them to use.
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 work on one more question, let’s work on 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 with no issue.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an item is the measurement of the total area that the object’s surface consist of. It is an essential part of the formula; thus, we must know how to find it.
There are a few distinctive ways to find the surface area of a prism. To calculate the surface area of a rectangular prism, you can use 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 work out 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 utilize 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 figure out the total surface area of a rectangular prism with the ensuing information.
l=8 in
b=5 in
h=7 in
To figure out this, we will replace these values 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 Calculating 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 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 will be able to calculate any prism’s volume and surface area. Try it out for yourself and see how simple it is!
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