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 made by taking a polygonal base and expanding its sides as far as it intersects the opposing base.
This article post will take you through what a prism is, its definition, different kinds, and the formulas for surface areas and volumes. We will also give examples of how to utilize the data provided.
What Is a Prism?
A prism is a 3D geometric figure with two congruent and parallel faces, well-known as bases, which take the form of a plane figure. The other faces are rectangles, and their amount rests 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.
The properties of a prism are interesting. The base and top both have an edge in parallel with the additional two sides, making them congruent to one another as well! This implies that every three dimensions - length and width in front and depth to the back - can be broken down into these four entities:
A lateral face (implying both height AND depth)
Two parallel planes which constitute of each base
An fictitious line standing upright across any given point on any side of this shape's core/midline—usually 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:
The rectangular prism is a regular kind of prism. It has six sides 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 faces. It appears almost like 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 total amount of space that an thing occupies. As an essential figure 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, where,
V = Volume
B = Base area
Finally, considering bases can have all types of figures, you are required to know a few formulas to figure out the surface area of the base. Despite that, we will touch upon that afterwards.
The Derivation of the Formula
To extract the formula for the volume of a rectangular prism, we need to observe 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
Right away, 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 refers to height, which is how thick our slice was.
Now that we have a formula for the volume of a rectangular prism, we can generalize it to any kind of prism.
Examples of How to Use the Formula
Now that we know the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s put them to use.
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=432 square inches
Now, consider one more problem, let’s figure out the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=450 cubic inches
As long as you possess the surface area and height, you will work out the volume without any issue.
The Surface Area of a Prism
Now, let’s talk 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 important part of the formula; thus, we must learn how to calculate 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 utilize 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 work out the surface area of a triangular prism, we will use this formula:
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 Finding the Surface Area of a Rectangular Prism
First, we will work on the total surface area of a rectangular prism with the following information.
To solve this, we will replace 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 compute the surface area of a triangular prism, we will find the total surface area by ensuing similar steps as earlier.
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)
SA = (40*7) + (2*60)
SA = 400 square inches
With this knowledge, you will be able to calculate any prism’s volume and surface area. Try it out for yourself and see how simple it is!
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