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Determine the area of a cirlce based on its perimeter

In this article I am going to show you how to determine the area of a circle when its perimeter is given.

To begin with let me first introduce the formulas for determining the perimeter and the area of a circle:

Perimeter
P
= Π * d or P = 2*Π *r

Area
A
= Π * r2

where d and r are the diameter and radius of the circle respectively.

As we can see that the only term that we need to know is the radius of the circle. So the first step is to isolate r from the formula for perimeter, which is shown below:

r = P / (2* Π )

by inserting r into the formula for area, we get:

A = Π * ( P / (2* Π ))2, which can be rewritten as: A = P2/(4 * Π )

Using this formula it is now possible to calculate the area of a circle when its perimeter is given.

To prove this is a valid formula, let's take a look at the following example as the final part of this article.

given that r = 3

P
= 3* Π * r = 6 * Π
A = Π * r2 = 9 * Π
A = P2/(4 * Π ) = (6 * Π )2 / (4 * Π ) = 36 * Π 2 / (4 * Π ) = 9 * Π

As you can see A = P2/(4 * Π ) gives the same result as the good old formula for area, this new formula can help you skip a lot of work in practice, since it is not always easy to measure the radius of a circle.
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