Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The figure’s name is derived from the fact that it is created by considering a polygonal base and stretching its sides till it cross the opposing base.
This article post will discuss what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also offer instances of how to use the details given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, which take the form of a plane figure. The other faces are rectangles, and their count depends 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 properties of a prism are interesting. The base and top each have an edge in common with the other two sides, creating them congruent to one another as well! This states that every three dimensions - length and width in front and depth to the back - can be decrypted into these four parts:
A lateral face (meaning both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any given point on any 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 meet
Types 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 resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism has two pentagonal bases and five rectangular faces. It seems 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 shape in geometry, the volume of a prism is very important for your studies.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Consequently, given that bases can have all kinds of shapes, you have to learn few formulas to calculate the surface area of the base. Despite that, we will touch upon that later.
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 sides that are all squares. The formula for the volume of a cube is V=s^3, assuming,
V = Volume
s = Side length
Immediately, we will get 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, which 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 kind of prism.
Examples of How to Utilize the Formula
Considering we understand the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s utilize these now.
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 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
Provided that you possess the surface area and height, you will work out the volume with no 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 occupies. It is an essential part of the formula; thus, we must learn how to calculate it.
There are a several varied ways to figure out the surface area of a prism. To measure 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 calculate the surface area of a triangular prism, we will employ 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 following information.
l=8 in
b=5 in
h=7 in
To figure out 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 following same 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. 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 knowledge, you will be able to work out any prism’s volume and surface area. Test it out for yourself and see how easy it is!
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