Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial figure in geometry. The shape’s name is originated from the fact that it is created by considering a polygonal base and extending its sides till it cross the opposing base.
This article post will talk about what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also provide instances of how to utilize the data given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, known as bases, which take the form of a plane figure. The additional faces are rectangles, and their amount depends 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 properties of a prism are astonishing. The base and top both have an edge in common with the additional two sides, creating them congruent to one another as well! This implies that all three dimensions - length and width in front and depth to the back - can be decrypted into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which make up each base
An illusory line standing upright across any given point on either 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
Kinds of Prisms
There are three primary types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular type of prism. It has six faces that are all rectangles. It resembles a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism has two pentagonal bases and five rectangular sides. It appears 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 measure of the total amount of area that an thing occupies. As an essential 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 kinds of shapes, you have to retain few formulas to figure out the surface area of the base. Despite that, we will go through that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we need to look at 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 take 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 implies 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 kind of prism.
Examples of How to Utilize the Formula
Now that we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s put them to use.
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, let’s try another question, 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
Provided that you possess the surface area and height, you will work out the volume with no issue.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an item is the measure of the total area that the object’s surface comprises of. It is an essential part of the formula; therefore, we must know how to calculate it.
There are a few varied 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), assuming,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To work out 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 Finding the Surface Area of a Rectangular Prism
First, we will work on the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To solve 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 Computing 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 same steps as before.
This prism consists of 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 knowledge, you should be able to figure out any prism’s volume and surface area. Check out for yourself and observe how simple it is!
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