Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The shape’s name is derived from the fact that it is made by taking into account a polygonal base and stretching its sides until it intersects the opposing 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 offer examples of how to employ the data given.
What Is a Prism?
A prism is a 3D 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 number relies 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 will be five sides.
Definition
The properties of a prism are fascinating. The base and top each have an edge in common with the other two sides, creating them congruent to one another as well! This means 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 constitute of each base
An imaginary line standing upright across any given point on either 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 join
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 sides that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It looks 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 calculation of the total amount of area that an object occupies. As an important 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
Consequently, since bases can have all types of figures, you are required to know a few formulas to figure out the surface area of the base. Despite that, we will go through that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we need to look at 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, assuming,
V = Volume
s = Side length
Right away, we will have a slice out of our cube that is h units thick. This slice will by itself be 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, 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 type of prism.
Examples of How to Use the Formula
Now that we have 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, consider another question, let’s figure out 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 calculate the volume with no issue.
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 occupies. It is an important part of the formula; thus, we must know how to find it.
There are a few distinctive methods 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), where,
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 utilize 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 Computing the Surface Area of a Rectangular Prism
Initially, we will work on the total surface area of a rectangular prism with the ensuing 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 Finding the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will find the total surface area by ensuing similar 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. 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 knowledge, you should be able to compute any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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