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 created by considering a polygonal base and extending its sides as far as it creates an equilibrium with the opposite base.
This article post will discuss what a prism is, its definition, different types, and the formulas for volume and surface area. We will also provide instances of how to use the data given.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The other faces are rectangles, and their count 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 parallel with the other two sides, making them congruent to each other as well! This means that all three dimensions - length and width in front and depth to the back - can be broken down into these four entities:
A lateral face (implying both height AND depth)
Two parallel planes which make up each base
An fictitious line standing upright through 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 primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common type of prism. It has six faces 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 looks 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 measurement of the total amount of space that an object occupies. As an important figure 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
Finally, considering bases can have all sorts of shapes, you have to know a few formulas to determine the surface area of the base. Still, we will touch upon that later.
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 three-dimensional object 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
Right away, we will get 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 refers to height, that is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can use it on any kind of prism.
Examples of How to Use the Formula
Now that we know the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, now let’s use them.
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
Considering 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 measure of the total area that the object’s surface consist of. It is an important part of the formula; thus, 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 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
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
Initially, we will figure out the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To figure out this, we will replace these values 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 Calculating the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will find 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. 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 data, you should be able to work out any prism’s volume and surface area. Test it out for yourself and see how easy it is!
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