Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is derived from the fact that it is made by taking into account a polygonal base and stretching its sides as far as it creates an equilibrium with the opposing base.
This blog post will take you through what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also offer instances of how to employ the information given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, known as bases, that take the shape of a plane figure. The additional faces are rectangles, and their number 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 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 each other 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 (implying both height AND depth)
Two parallel planes which constitute of each base
An illusory 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 meet
Kinds of Prisms
There are three major types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular kind 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 faces. It seems 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 sum of space that an object occupies. As an crucial figure 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
Finally, considering bases can have all kinds of figures, you are required to know a few formulas to calculate the surface area of the base. Despite that, we will go through that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we have to observe a cube. A cube is a three-dimensional 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 have 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 refers to height, which 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 kind of prism.
Examples of How to Use the Formula
Considering we have the formulas for the volume of a rectangular prism, triangular 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, consider 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
Provided that 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 about the surface area. The surface area of an object is the measure of the total area that the object’s surface consist of. It is an important part of the formula; consequently, we must understand how to find it.
There are a few different methods to find the surface area of a prism. To calculate 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 figure out 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 Finding the Surface Area of a Rectangular Prism
Initially, we will work on the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To figure out this, we will plug 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 Finding 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 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. 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 should be able to calculate any prism’s volume and surface area. Check out for yourself and observe how easy it is!
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