Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is a vital figure in geometry. The figure’s name is originated from the fact that it is made by taking a polygonal base and extending its sides until it intersects 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 data given.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, well-known as bases, that take the form of a plane figure. The additional 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 would be five sides.
Definition
The characteristics of a prism are fascinating. 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 every three dimensions - length and width in front and depth to the back - can be decrypted into these four parts:
A lateral face (implying both height AND depth)
Two parallel planes which constitute of each base
An imaginary line standing upright through any provided 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 meet
Types of Prisms
There are three main types 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 close to a triangular prism, but the pentagonal shape of the base sets it apart.
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 relevant in your learning.
The formula for the volume of a rectangular prism is V=B*h, where,
V = Volume
B = Base area
h= Height
Consequently, given that bases can have all types of shapes, you have to retain few formulas to figure out the surface area of the base. Still, we will go through that afterwards.
The Derivation of the Formula
To obtain the formula for the volume of a rectangular prism, we need to look at a cube. A cube is a 3D item 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
Immediately, we will take a slice out of our cube that is h units thick. This slice will make 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 kind of prism.
Examples of How to Use the Formula
Considering we know 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 work on one more 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
Considering 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 discuss about 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 essential part of the formula; thus, we must learn how to find it.
There are a few distinctive methods to figure out the surface area of a prism. To calculate the surface area of a rectangular prism, you can use 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 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 Computing the Surface Area of a Rectangular Prism
First, we will determine the total surface area of a rectangular prism with the following dimensions.
l=8 in
b=5 in
h=7 in
To figure out 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 calculate the surface area of a triangular prism, we will figure out the total surface area by following similar 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. 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 information, you should be able to work out any prism’s volume and surface area. Test it out for yourself and observe how easy it is!
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