Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is derived from the fact that it is made by taking a polygonal base and stretching its sides as far as it creates an equilibrium with the opposing base.
This blog 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 examples of how to employ the information provided.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, which take the shape of a plane figure. The additional faces are rectangles, and their amount rests 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 each have an edge in parallel with the additional two sides, creating them congruent to one another as well! This states 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 make up each base
An fictitious line standing upright through any provided point on any side of this figure'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 main kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common kind 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 consists of two pentagonal bases and five rectangular sides. It appears close to a triangular prism, but the pentagonal shape of the base makes it apart.
The Formula for the Volume of a Prism
Volume is a measure of the sum of area that an thing occupies. As an crucial figure in geometry, the volume of a prism is very important for your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all sorts of shapes, you will need to retain few formulas to determine the surface area of the base. Despite that, we will touch upon that afterwards.
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 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 get 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 refers to the base area of the rectangle. The h in the formula implies the height, that 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 Use 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 work on 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, consider 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
As long as you have the surface area and height, you will figure out the volume with no problem.
The Surface Area of a Prism
Now, let’s discuss regarding the surface area. The surface area of an item is the measure of the total area that the object’s surface consist of. It is an essential part of the formula; thus, we must learn how to calculate it.
There are a several varied ways 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 compute the surface area of a triangular prism, we will use 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
Initially, we will determine the total surface area of a rectangular prism with the following data.
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 Finding the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will work on the total surface area by ensuing similar 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. 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 knowledge, you should 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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