Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The shape’s name is originated from the fact that it is made by considering a polygonal base and extending its sides until it creates an equilibrium with the opposite base.
This article post will discuss what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also give instances of how to utilize the data provided.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, called bases, that take the shape of a plane figure. The other faces are rectangles, and their number 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 properties of a prism are astonishing. The base and top each have an edge in common with the additional two sides, creating them congruent to one another as well! This implies that all 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 make up each base
An illusory line standing upright across any provided point on any side of this shape's core/midline—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Kinds of Prisms
There are three main types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type of prism. It has six sides that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism has 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 measure of the sum of space that an thing occupies. As an crucial 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, where,
V = Volume
B = Base area
h= Height
Ultimately, given that bases can have all types of figures, you have to learn few formulas to figure out the surface area of the base. Despite that, we will touch upon that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we have to look at a cube. A cube is a 3D item 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
Immediately, 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 implies 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 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 work on one more problem, 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
Considering that you have the surface area and height, you will calculate the volume with no problem.
The Surface Area of a Prism
Now, let’s discuss regarding 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 important part of the formula; therefore, we must learn how to find it.
There are a several different ways to figure out the surface area of a prism. To figure out the surface area of a rectangular prism, you can use 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 figure out the surface area of a triangular prism, we will use 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 data.
l=8 in
b=5 in
h=7 in
To figure out this, we will plug 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 calculate the surface area of a triangular prism, we will work on the total surface area by ensuing identical steps as before.
This prism consists of 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 information, you will be able to work out any prism’s volume and surface area. Test it out for yourself and see how simple it is!
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