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 into account a polygonal base and extending its sides until it cross the opposite base.
This blog post will discuss what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also provide instances of how to use the details given.
What Is a Prism?
A prism is a 3D geometric shape with two congruent and parallel faces, well-known as bases, that take the form of a plane figure. The additional faces are rectangles, and their count depends on how many sides the identical base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there will be five sides.
Definition
The characteristics of a prism are fascinating. The base and top both have an edge in common with the additional two sides, making 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 deconstructed into these four parts:
A lateral face (implying both height AND depth)
Two parallel planes which make up each base
An fictitious line standing upright through any provided point on any side of this shape's core/midline—also known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Types 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 faces that are all rectangles. It matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism has two pentagonal bases and five rectangular sides. It looks close to a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a calculation of the total amount of area that an thing occupies. As an important 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
Finally, considering bases can have all sorts of figures, you have to learn few formulas to determine the surface area of the base. Despite that, we will go through that later.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we are required 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, where,
V = Volume
s = Side length
Immediately, 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 type of prism.
Examples of How to Utilize the Formula
Considering we understand the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, now let’s use them.
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 one more 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
As long as you possess the surface area and height, you will calculate the volume with no 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 comprises of. It is an important part of the formula; therefore, we must understand how to calculate it.
There are a several varied ways to work out the surface area of a prism. To figure out the surface area of a rectangular prism, you can use 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 utilize 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 Calculating the Surface Area of a Rectangular Prism
Initially, 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 calculate 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 calculate the surface area of a triangular prism, we will figure out the total surface area by following same steps as priorly used.
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 data, you will be able to figure out any prism’s volume and surface area. Try it out for yourself and observe how simple it is!
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