Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The figure’s name is originated from the fact that it is made by taking a polygonal base and stretching its sides as far as it cross the opposing base.
This blog 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 utilize the details provided.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, called bases, that take the shape of a plane figure. The other faces are rectangles, and their number rests 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 astonishing. The base and top both have an edge in parallel with the additional two sides, creating them congruent to one another as well! This states that all three dimensions - length and width in front and depth to the back - can be decrypted into these four entities:
A lateral face (meaning both height AND depth)
Two parallel planes which make up each base
An fictitious line standing upright across any provided point on either 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 join
Kinds of Prisms
There are three primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type 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 consists of two pentagonal bases and five rectangular faces. It looks a lot like a triangular prism, but the pentagonal shape of the base sets it apart.
The Formula for the Volume of a Prism
Volume is a measurement of the sum of space that an object occupies. As an important 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, assuming,
V = Volume
B = Base area
h= Height
Ultimately, considering bases can have all kinds of figures, you have to retain few formulas to figure out 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 need to observe 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, where,
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 refers to the base area of the rectangle. The h in the formula stands for 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 kind of prism.
Examples of How to Use the Formula
Since we understand the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s put them to use.
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 problem, let’s calculate 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 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 important part of the formula; thus, we must learn how to find it.
There are a several different methods to find the surface area of a prism. To measure the surface area of a rectangular prism, you can employ 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
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 Computing 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 replace these numbers 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 find the total surface area by ensuing similar steps as earlier.
This prism will have 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 knowledge, you should be able to figure out any prism’s volume and surface area. Try it out for yourself and see how simple it is!
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