Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The shape’s name is derived from the fact that it is made by considering a polygonal base and stretching its sides till it cross 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 provide instances of how to utilize the information provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, that take the form of a plane figure. The other faces are rectangles, and their count 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 properties of a prism are interesting. The base and top both have an edge in parallel with the additional two sides, creating 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 (meaning both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright across 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
Kinds of Prisms
There are three primary kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular 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 faces.
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 measurement of the total amount of area that an item occupies. As an essential 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
Ultimately, since bases can have all kinds of shapes, you will need to know a few formulas to calculate the surface area of the base. Despite that, we will touch upon that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional item 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 have 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 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 kind of prism.
Examples of How to Utilize the Formula
Considering we know the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s put them to use.
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 problem, 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
Considering that you possess the surface area and height, you will figure out the volume without any issue.
The Surface Area of a Prism
Now, let’s discuss regarding the surface area. The surface area of an object is the measurement of the total area that the object’s surface consist of. It is an crucial part of the formula; therefore, we must know how to find it.
There are a several distinctive ways 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
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
First, we will figure out the total surface area of a rectangular prism with the following data.
l=8 in
b=5 in
h=7 in
To figure out this, we will put 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 Calculating 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 ensuing identical steps as before.
This prism consists of 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 figure out any prism’s volume and surface area. Check out for yourself and observe how easy it is!
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