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 made by taking into account a polygonal base and expanding its sides till it creates an equilibrium with the opposite base.
This blog post will discuss what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also provide examples of how to utilize the information provided.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The other faces are rectangles, and their amount relies 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 would be five sides.
Definition
The properties of a prism are interesting. The base and top both have an edge in common with the other 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 imaginary line standing upright across any given point on any side of this figure'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 main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common type of prism. It has six sides that are all rectangles. It looks like a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism comprises of two pentagonal bases and five rectangular faces. 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 crucial shape 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
Finally, given that bases can have all sorts of shapes, you have to know a few formulas to determine the surface area of the base. However, we will go through that afterwards.
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
Right away, 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 dense 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
Now that we understand the formulas for the volume of a triangular prism, rectangular prism, and pentagonal prism, let’s utilize these now.
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 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
Considering that you have the surface area and height, you will figure out the volume with no issue.
The Surface Area of a Prism
Now, let’s talk about the surface area. The surface area of an item is the measure of the total area that the object’s surface occupies. It is an important part of the formula; therefore, we must know how to find it.
There are a several distinctive ways to figure out 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 calculate 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 Calculating the Surface Area of a Rectangular Prism
First, we will figure out the total surface area of a rectangular prism with the ensuing information.
l=8 in
b=5 in
h=7 in
To solve 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 Computing 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 identical 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. Thus,
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 should be able to work out any prism’s volume and surface area. Check out for yourself and see how simple it is!
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