Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The figure’s name is originated from the fact that it is made by taking a polygonal base and expanding its sides till it intersects the opposing base.
This blog post will take you through what a prism is, its definition, different types, and the formulas for volume and surface area. We will also provide instances of how to utilize the data given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, known as bases, that take the form of a plane figure. The additional faces are rectangles, and their count relies on how many sides the similar 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 astonishing. The base and top both have an edge in parallel with the additional two sides, making them congruent to one another as well! This implies that every three dimensions - length and width in front and depth to the back - can be decrypted 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 across any given point on any 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 primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common kind of prism. It has six sides that are all rectangles. It matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism comprises of two pentagonal bases and five rectangular faces. It looks close to a triangular prism, but the pentagonal shape of the base makes 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 crucial 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, assuming,
V = Volume
B = Base area
h= Height
Finally, considering bases can have all kinds of shapes, you have to learn few formulas to calculate the surface area of the base. Still, we will touch upon that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we have to look at a cube. A cube is a 3D object with six faces that are all squares. The formula for the volume of a cube is V=s^3, assuming,
V = Volume
s = Side length
Now, 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 stands for the base area of the rectangle. The h in the formula stands for height, that 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 Utilize the Formula
Since we have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, now let’s use them.
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, let’s work on another question, 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 possess the surface area and height, you will figure out the volume with no issue.
The Surface Area of a Prism
Now, let’s discuss about the surface area. The surface area of an item is the measure of the total area that the object’s surface occupies. It is an essential part of the formula; consequently, we must know how to find it.
There are a several different ways to work out the surface area of a prism. To calculate 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 compute 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 utilize 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 ensuing dimensions.
l=8 in
b=5 in
h=7 in
To calculate this, we will plug 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 figure out the total surface area by ensuing similar 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. 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 data, you will be able to figure out any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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