Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The figure’s name is originated from the fact that it is made by taking a polygonal base and stretching its sides as far as it cross the opposite base.
This article 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 information provided.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, which take the shape of a plane figure. The additional 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 would be five sides.
Definition
The characteristics of a prism are interesting. The base and top each have an edge in common with the additional two sides, creating them congruent to each other as well! This states that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four parts:
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 either side of this figure's core/midline—also 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 regular type 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 has two pentagonal bases and five rectangular sides. It appears a lot like a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a measurement of the total amount of area that an thing occupies. As an essential figure 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
Ultimately, given that bases can have all types of figures, you are required to retain few formulas to determine the surface area of the base. However, we will go through that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we are required to observe 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
Immediately, 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, which 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
Considering we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s utilize these now.
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, consider one more problem, 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
Considering that you possess the surface area and height, you will calculate the volume with no problem.
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 comprises of. It is an crucial part of the formula; thus, we must know how to calculate it.
There are a several varied methods to figure out the surface area of a prism. To figure out 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 work 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
First, we will determine the total surface area of a rectangular prism with the ensuing data.
l=8 in
b=5 in
h=7 in
To calculate this, we will put 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 Computing the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will find 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. Hence,
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 work out any prism’s volume and surface area. Try it out for yourself and observe how easy it is!
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