Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The shape’s name is originated from the fact that it is made by taking into account a polygonal base and stretching its sides as far as it creates an equilibrium with 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 provide examples of how to use the information provided.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, known as bases, which take the shape of a plane figure. The additional faces are rectangles, and their amount depends 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 will be five sides.
Definition
The characteristics of a prism are interesting. The base and top both have an edge in parallel with the other two sides, creating them congruent to one another as well! This states that every three dimensions - length and width in front and depth to the back - can be deconstructed into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which constitute of each base
An imaginary line standing upright across any provided point on any 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 join
Types of Prisms
There are three main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common kind of prism. It has six sides that are all rectangles. It matches the looks of 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 almost like a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a measurement of the sum of space that an item occupies. As an important figure 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 are required to retain few formulas to figure out the surface area of the base. Still, we will touch upon 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 item 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 have 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 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 Utilize the Formula
Considering we know the formulas for the volume of a pentagonal prism, triangular prism, and rectangular 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 work on 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 problem.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an object is the measure of the total area that the object’s surface consist of. It is an essential part of the formula; consequently, we must learn how to calculate it.
There are a several different 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 employ 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 Finding the Surface Area of a Rectangular Prism
First, 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 calculate this, we will put these numbers into the corresponding 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 find the surface area of a triangular prism, we will find the total surface area by following same steps as priorly used.
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 see how simple it is!
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