Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The figure’s name is derived from the fact that it is created by taking into account a polygonal base and stretching its sides as far as it intersects the opposing base.
This blog post will talk about what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also give examples of how to use the information given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, well-known as bases, that take the form 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 astonishing. The base and top both have an edge in common with the other 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 entities:
A lateral face (meaning both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright through any provided point on any side of this shape's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
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 looks like 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 almost like 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 sum of space that an object occupies. As an important 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 figures, you will need to learn few formulas to determine the surface area of the base. However, 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 3D object with six sides that are all squares. The formula for the volume of a cube is V=s^3, assuming,
V = Volume
s = Side length
Immediately, we will take 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 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 have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular 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 work on another question, 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 calculate the volume with no 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 comprises of. It is an essential part of the formula; consequently, we must know how to calculate it.
There are a several different methods to find the surface area of a prism. To measure the surface area of a rectangular prism, you can utilize this: A=2(lb + bh + lh), assuming,
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 utilize this formula:
SA=(S1+S2+S3)L+bh
assuming,
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 information.
l=8 in
b=5 in
h=7 in
To figure out this, we will put 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 Calculating the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will figure out the total surface area by ensuing identical steps as earlier.
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. Test it out for yourself and see how simple it is!
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