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 created by taking a polygonal base and extending its sides until it creates an equilibrium with the opposing base.
This article post will talk about what a prism is, its definition, different types, and the formulas for surface areas and volumes. We will also give examples of how to employ the details given.
What Is a Prism?
A prism is a three-dimensional geometric shape with two congruent and parallel faces, called bases, that take the shape of a plane figure. The other faces are rectangles, and their amount depends 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 characteristics of a prism are astonishing. The base and top each have an edge in parallel with the other two sides, creating them congruent to each other as well! This means that every three dimensions - length and width in front and depth to the back - can be decrypted into these four entities:
A lateral face (signifying both height AND depth)
Two parallel planes which make up each base
An imaginary line standing upright through any given 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 join
Types of Prisms
There are three major types 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 faces.
The pentagonal prism has two pentagonal bases and five rectangular sides. It seems 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 calculation of the total amount of space that an thing occupies. As an crucial shape in geometry, the volume of a prism is very relevant in your studies.
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 kinds of shapes, you are required to know a few formulas to calculate the surface area of the base. Despite that, we will touch upon that later.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we are required 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, where,
V = Volume
s = Side length
Immediately, we will get 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 implies the base area of the rectangle. The h in the formula implies the height, which 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 have the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, now let’s use them.
First, let’s figure out 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 possess the surface area and height, you will figure out the volume with no problem.
The Surface Area of a Prism
Now, let’s discuss about the surface area. The surface area of an item is the measurement of the total area that the object’s surface occupies. It is an essential part of the formula; consequently, we must know how to find it.
There are a few different ways to find the surface area of a prism. To figure out 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 calculate the surface area of a triangular prism, we will utilize 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 ensuing dimensions.
l=8 in
b=5 in
h=7 in
To figure out 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 Calculating 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 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. 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 knowledge, you should be able to work out any prism’s volume and surface area. Check out for yourself and observe how easy it is!
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