Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a crucial shape in geometry. The figure’s name is derived from the fact that it is created by taking a polygonal base and extending its sides till it creates an equilibrium with the opposite base.
This blog post will discuss what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also take you through some instances of how to utilize 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, which take the shape of a plane figure. The additional faces are rectangles, and their amount relies on how many sides the similar base has. For instance, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there would be five sides.
Definition
The properties of a prism are astonishing. The base and top each have an edge in parallel with the other two sides, making 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 parts:
A lateral face (implying both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright through any given point on any side of this figure'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 types of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a regular kind of prism. It has six faces that are all rectangles. It matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism comprises of two pentagonal bases and five rectangular sides. It seems 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 measure of the sum of space that an thing occupies. As an important 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, assuming,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all types of figures, you have to know a few formulas to figure out the surface area of the base. Still, we will go through that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we are required to look at a cube. A cube is a three-dimensional item 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
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 stands for the base area of the rectangle. The h in the formula refers to 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
Since we understand the formulas for the volume of a rectangular prism, triangular prism, and pentagonal 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, let’s try one more 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
Considering that you have the surface area and height, you will figure out the volume with no issue.
The Surface Area of a Prism
Now, let’s discuss 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 crucial 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 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 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 utilize 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 figure out 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 put these values 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 compute the surface area of a triangular prism, we will work on the total surface area by ensuing similar steps as before.
This prism will have 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 data, 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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