Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The figure’s name is derived from the fact that it is created by taking into account a polygonal base and expanding its sides as far as it intersects 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 offer examples of how to utilize the details provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, known as bases, that take the form of a plane figure. The additional faces are rectangles, and their amount relies 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 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 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 constitute of each base
An illusory line standing upright across any given point on any side of this figure's core/midline—also 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 kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common 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 sides.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It seems a lot like 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 sum of area that an item occupies. As an essential figure in geometry, the volume of a prism is very relevant in your learning.
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 sorts of figures, you have to know a few formulas to figure out the surface area of the base. However, we will go through that afterwards.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we need to look at 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
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 stands for the base area of the rectangle. The h in the formula stands for height, which 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 type of prism.
Examples of How to Utilize the Formula
Now that we know the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, now let’s use them.
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 one more problem, 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
As long as you have 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 consist of. It is an important part of the formula; consequently, we must understand how to calculate it.
There are a few distinctive methods to find the surface area of a prism. To measure 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 figure out 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 utilize 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 information.
l=8 in
b=5 in
h=7 in
To solve this, we will plug 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 Computing the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will find the total surface area by following similar 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. 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 calculate any prism’s volume and surface area. Test it out for yourself and see how easy it is!
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