Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is originated from the fact that it is made by taking 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 surface areas and volumes. We will also offer examples of how to use the data given.
What Is a Prism?
A prism is a three-dimensional 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 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 will be five sides.
Definition
The characteristics of a prism are interesting. The base and top each have an edge in parallel with the additional two sides, making them congruent to each other as well! This means that all three dimensions - length and width in front and depth to the back - can be broken down into these four parts:
A lateral face (meaning both height AND depth)
Two parallel planes which constitute of each base
An imaginary line standing upright across any provided point on either side of this figure's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
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 sides that are all rectangles. It looks like a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism consists of two pentagonal bases and five rectangular faces. It appears 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 calculation of the sum of space that an thing occupies. As an essential shape in geometry, the volume of a prism is very important for your learning.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Finally, given that bases can have all types of figures, you have to retain few formulas to calculate the surface area of the base. Still, we will go through that afterwards.
The Derivation of the Formula
To extract 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, assuming,
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 stands for the base area of the rectangle. The h in the formula stands for height, which is how thick 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 rectangular prism, triangular 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 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 have the surface area and height, you will figure 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 essential part of the formula; thus, we must know how to find it.
There are a few different ways 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), 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 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 work on 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 plug 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 find the surface area of a triangular prism, we will work on the total surface area by ensuing similar steps as earlier.
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 data, you will be able to calculate any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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