Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The shape’s name is originated from the fact that it is created by taking into account a polygonal base and extending its sides till it intersects the opposing base.
This blog post will discuss what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also provide instances of how to employ the details provided.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, well-known as bases, that take the shape of a plane figure. The other faces are rectangles, and their number 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 fascinating. The base and top both have an edge in common with the additional two sides, creating them congruent to one another as well! This means that every three dimensions - length and width in front and depth to the back - can be deconstructed 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 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 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 sides that are all rectangles. It looks like a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It appears almost like a triangular prism, but the pentagonal shape of the base makes it apart.
The Formula for the Volume of a Prism
Volume is a measurement of the total amount of area that an item occupies. As an essential 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, considering bases can have all types of figures, you have to know a few formulas to calculate the surface area of the base. Still, we will touch upon that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we have to observe a cube. A cube is a three-dimensional item 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
Right away, we will have a slice out of our cube that is h units thick. This slice will create 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 generalize it to any kind of prism.
Examples of How to Utilize the Formula
Since we know the formulas for the volume of a triangular prism, rectangular 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 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
Provided that you have 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 item is the measurement of the total area that the object’s surface consist of. It is an important part of the formula; consequently, we must understand how to calculate it.
There are a several 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 compute 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 utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding the Surface Area of a Rectangular Prism
Initially, we will determine the total surface area of a rectangular prism with the ensuing data.
l=8 in
b=5 in
h=7 in
To solve this, we will plug 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 Finding the Surface Area of a Triangular Prism
To compute the surface area of a triangular prism, we will find the total surface area by ensuing identical 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 will be able to work out any prism’s volume and surface area. Try it out for yourself and observe how easy it is!
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