Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital 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 intersects the opposing base.
This blog post will take you through what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also offer examples of how to use the details provided.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, which take the shape of a plane figure. The additional faces are rectangles, and their amount rests 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 characteristics of a prism are astonishing. The base and top both have an edge in common with the other two sides, creating them congruent to each other 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 illusory line standing upright through any given point on any side of this shape's core/midline—also known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Types of Prisms
There are three primary kinds 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 sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It looks almost 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 measure of the total amount of space that an item occupies. As an important shape 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 kinds of shapes, you will need to know a few formulas to calculate the surface area of the base. Despite that, we will touch upon that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we are required to observe a cube. A cube is a three-dimensional item 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 implies 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 know the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s put them to use.
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 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
Considering that you have the surface area and height, you will calculate the volume with no problem.
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 consist of. It is an important part of the formula; therefore, we must learn how to calculate it.
There are a several varied methods to figure 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 compute the surface area of a triangular prism, we will use this formula:
SA=(S1+S2+S3)L+bh
where,
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
First, we will determine the total surface area of a rectangular prism with the ensuing data.
l=8 in
b=5 in
h=7 in
To figure out this, we will replace 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 Finding the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will find the total surface area by ensuing same steps as priorly used.
This prism consists of 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 knowledge, you will be able to figure out any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
Use Grade Potential to Enhance Your Arithmetics Skills Now
If you're struggling to understand prisms (or any other math concept, contemplate signing up for a tutoring session with Grade Potential. One of our expert teachers can assist you study the [[materialtopic]187] so you can nail your next test.
