Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital shape in geometry. The figure’s name is originated from the fact that it is created by taking a polygonal base and stretching its sides as far as it cross the opposing base.
This blog post will take you through what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also take you through some examples of how to use the details given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, called bases, which take the form of a plane figure. The other faces are rectangles, and their count depends on how many sides the identical 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 characteristics of a prism are astonishing. The base and top both have an edge in parallel with the other two sides, creating them congruent to each other as well! This states that all three dimensions - length and width in front and depth to the back - can be decrypted into these four parts:
A lateral face (meaning both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any provided point on any side of this shape's core/midline—known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes meet
Kinds of Prisms
There are three primary 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 matches the looks of a box.
The triangular prism has two triangular bases and three rectangular faces.
The pentagonal prism has two pentagonal bases and five rectangular faces. It appears a lot 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 space that an thing occupies. As an crucial shape 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
Ultimately, since bases can have all types of shapes, you have to know a few formulas to determine the surface area of the base. However, we will touch upon that later.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we have to look at 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
Now, 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 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 type of prism.
Examples of How to Use the Formula
Since we have the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, now let’s use them.
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 one more problem, 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 possess the surface area and height, you will calculate the volume without any 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 comprises of. It is an essential part of the formula; thus, we must know how to find it.
There are a few varied ways to work out the surface area of a prism. To measure the surface area of a rectangular prism, you can employ this: A=2(lb + bh + lh), assuming,
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
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
Initially, we will figure out 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 plug 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 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. 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 information, you will be able to work out any prism’s volume and surface area. Test it out for yourself and observe how simple it is!
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