Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The shape’s name is originated from the fact that it is made by taking a polygonal base and stretching its sides as far as it intersects the opposite base.
This article post will take you through what a prism is, its definition, different types, 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 3D geometric figure with two congruent and parallel faces, well-known as bases, that take the shape of a plane figure. The other faces are rectangles, and their number relies 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 would be five sides.
Definition
The characteristics of a prism are fascinating. The base and top both have an edge in common with the other two sides, making them congruent to one another as well! This implies that all three dimensions - length and width in front and depth to the back - can be deconstructed into these four parts:
A lateral face (meaning both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright across any given point on either side of this shape'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 types 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 matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism has two pentagonal bases and five rectangular faces. It appears a lot like 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 space that an item occupies. As an essential 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, assuming,
V = Volume
B = Base area
h= Height
Finally, given that bases can have all sorts of shapes, you are required to know a few formulas to calculate the surface area of the base. However, we will go through that afterwards.
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, where,
V = Volume
s = Side length
Now, 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 use it on any type of prism.
Examples of How to Use the Formula
Now that we have the formulas for the volume of a rectangular prism, triangular 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, consider one more question, let’s work on 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 possess the surface area and height, you will calculate 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 measurement of the total area that the object’s surface occupies. It is an crucial part of the formula; consequently, we must understand how to calculate 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 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 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 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 work on the total surface area of a rectangular prism with the following information.
l=8 in
b=5 in
h=7 in
To solve this, we will plug these values 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 Calculating the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will figure out the total surface area by following 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. 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 will be able to work out any prism’s volume and surface area. Check out for yourself and see how simple it is!
Use Grade Potential to Better Your Arithmetics Abilities Now
If you're have a tough time understanding prisms (or any other math subject, contemplate signing up for a tutoring class with Grade Potential. One of our experienced tutors can help you learn the [[materialtopic]187] so you can ace your next examination.
