Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The figure’s name is originated from the fact that it is made by taking a polygonal base and extending its sides as far as it intersects the opposite base.
This blog post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also give examples of how to utilize the data given.
What Is a Prism?
A prism is a three-dimensional 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 amount rests 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 characteristics of a prism are astonishing. The base and top both have an edge in common with the additional two sides, making them congruent to each other as well! This implies that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
A lateral face (meaning both height AND depth)
Two parallel planes which constitute of each base
An illusory line standing upright through any provided point on either 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 meet
Types of Prisms
There are three major types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular kind of prism. It has six sides 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 comprises of two pentagonal bases and five rectangular sides. It seems close to 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 sum of area that an item occupies. As an essential figure in geometry, the volume of a prism is very relevant in your learning.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Ultimately, given that bases can have all kinds of figures, you have to learn few formulas to calculate the surface area of the base. Still, we will go through that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we have to observe a cube. A cube is a 3D object 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
Right away, 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 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 Use the Formula
Since we know 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 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
As long as 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 talk about the surface area. The surface area of an object is the measurement of the total area that the object’s surface comprises of. It is an essential part of the formula; consequently, we must know how to calculate it.
There are a few varied methods to find the surface area of a prism. To measure the surface area of a rectangular prism, you can employ 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 figure out 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 Finding the Surface Area of a Rectangular Prism
Initially, we will figure out the total surface area of a rectangular prism with the following 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 Computing the Surface Area of a Triangular Prism
To calculate the surface area of a triangular prism, we will figure out the total surface area by following similar steps as before.
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 information, you will be able to figure out any prism’s volume and surface area. Check out for yourself and observe how simple it is!
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