Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is originated from the fact that it is created by taking a polygonal base and stretching its sides till it creates an equilibrium with the opposing base.
This article post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also provide instances of how to use the data provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, well-known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their count rests on how many sides the similar base has. For instance, 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 astonishing. The base and top each have an edge in parallel with the other two sides, creating them congruent to each other as well! This means that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four parts:
A lateral face (implying both height AND depth)
Two parallel planes which make up each base
An fictitious line standing upright across any provided point on any side of this figure'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 kind of prism. It has six faces that are all rectangles. It looks like 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 appears a lot like a triangular prism, but the pentagonal shape of the base sets it apart.
The Formula for the Volume of a Prism
Volume is a measurement of the total amount of space that an thing occupies. As an crucial shape 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
Finally, considering bases can have all types of shapes, you will need to learn few formulas to determine the surface area of the base. Still, we will touch upon that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we are required 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, where,
V = Volume
s = Side length
Immediately, we will take 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 implies the base area of the rectangle. The h in the formula stands for height, which is how dense 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
Considering we know the formulas for the volume of a pentagonal prism, triangular prism, and rectangular 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 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
As long as you have the surface area and height, you will figure out the volume without any issue.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an object is the measure of the total area that the object’s surface comprises of. It is an essential part of the formula; consequently, we must understand how to find it.
There are a several distinctive methods to work out the surface area of a prism. To measure 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 calculate 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 utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding the Surface Area of a Rectangular Prism
First, 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 replace 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 find the surface area of a triangular prism, we will find the total surface area by ensuing similar 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. 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 calculate any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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