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 considering a polygonal base and extending its sides until it cross the opposing base.
This article post will take you through what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also offer instances of how to use the details provided.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, called bases, which 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 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 common with the other two sides, creating 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 parts:
A lateral face (meaning both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright through 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 meet
Kinds of Prisms
There are three major types 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 faces.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. 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 area that an object occupies. As an crucial shape in geometry, the volume of a prism is very relevant in your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Finally, given that bases can have all kinds of figures, you have to learn few formulas to calculate the surface area of the base. Still, we will touch upon that afterwards.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to observe 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 get 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 stands for the base area of the rectangle. The h in the formula implies the height, that 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 Utilize the Formula
Now that we understand the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s utilize these now.
First, let’s work on 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 figure out 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 work out the volume with no problem.
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 comprises of. It is an crucial part of the formula; thus, we must understand how to find it.
There are a few different methods to figure out the surface area of a prism. To measure the surface area of a rectangular prism, you can employ this: A=2(lb + bh + lh), assuming,
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
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 use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Calculating the Surface Area of a Rectangular Prism
Initially, we will work on the total surface area of a rectangular prism with the ensuing data.
l=8 in
b=5 in
h=7 in
To calculate this, we will put these values into the corresponding 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 calculate the surface area of a triangular prism, we will figure out the total surface area by following identical 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. 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 information, you should be able to work out any prism’s volume and surface area. Try it out for yourself and see how easy it is!
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