Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The shape’s name is derived from the fact that it is created by considering a polygonal base and expanding its sides till it creates an equilibrium with the opposing 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 offer examples of how to employ the details provided.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, well-known as bases, which take the form of a plane figure. The additional faces are rectangles, and their number rests on how many sides the identical base has. For instance, 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 other two sides, making 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 deconstructed into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which constitute of each base
An fictitious 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 main 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 resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism has two pentagonal bases and five rectangular faces. It seems 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 space that an object occupies. As an essential shape in geometry, the volume of a prism is very important for your studies.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Ultimately, considering bases can have all types of figures, you have to learn few formulas to figure out 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 are required to observe 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
Right away, we will take 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 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 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 calculate 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
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 talk 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 crucial part of the formula; consequently, we must know how to calculate it.
There are a several distinctive ways to find the surface area of a prism. To figure out 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 figure 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 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 figure out the total surface area of a rectangular prism with the following data.
l=8 in
b=5 in
h=7 in
To solve this, we will replace these numbers 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 Computing 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 following same 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. 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 calculate any prism’s volume and surface area. Check out for yourself and see how easy it is!
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