Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The shape’s name is derived from the fact that it is created by taking into account a polygonal base and stretching its sides until it intersects the opposing base.
This article post will take you through what a prism is, its definition, different kinds, and the formulas for volume and surface area. We will also offer instances of how to utilize 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 additional 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 properties of a prism are fascinating. The base and top each have an edge in parallel with the additional two sides, creating 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 decrypted into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any provided point on either 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 common 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 faces.
The pentagonal prism has two pentagonal bases and five rectangular sides. It appears close to 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 total amount of area that an object occupies. As an crucial figure in geometry, the volume of a prism is very relevant in 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 are required to know a few formulas to determine the surface area of the base. Still, 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 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, 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 stands for the base area of the rectangle. The h in the formula implies the height, which is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any kind of prism.
Examples of How to Use the Formula
Considering we have the formulas for the volume of a pentagonal prism, triangular prism, and rectangular prism, let’s put them to use.
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 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
Considering that you possess the surface area and height, you will figure out the volume with no problem.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an item is the measurement of the total area that the object’s surface consist of. It is an important part of the formula; thus, we must learn how to calculate it.
There are a few varied methods to find the surface area of a prism. To calculate the surface area of a rectangular prism, you can employ 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 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 replace 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 find the total surface area by following same 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. 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 compute any prism’s volume and surface area. Check out for yourself and see how easy it is!
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