Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important figure in geometry. The shape’s name is originated from the fact that it is created by taking into account a polygonal base and extending its sides until it intersects the opposite base.
This blog post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also give examples of how to use the details given.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, known as bases, which 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 will 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, making them congruent to each other as well! This implies that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright through any provided 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 main types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common type of prism. It has six faces that are all rectangles. It looks like 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 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 calculation of the sum of area that an thing occupies. As an essential 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
Ultimately, considering bases can have all kinds of shapes, you have to retain few formulas to determine the surface area of the base. However, 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 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, where,
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 refers to the base area of the rectangle. The h in the formula implies the 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 type of prism.
Examples of How to Utilize the Formula
Considering we know the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, now let’s use them.
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 try one more question, 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
Considering that 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 talk about the surface area. The surface area of an object is the measure of the total area that the object’s surface comprises of. It is an important part of the formula; therefore, we must understand how to find it.
There are a several distinctive methods to work out the surface area of a prism. To figure out the surface area of a rectangular prism, you can utilize 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 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 use SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding 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 work on the total surface area by following identical 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. 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 information, you will be able to calculate any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
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