The formula to find the area of any regular polygon is (While a symbolizes apothegm and p symbolizes perimeter.) ap/2
NOTE: An apothegm is basically the radius of a regular polygon, in that it is half of the height. Please do not confuse the radius and apothegm of a regular polygon, as they are two different things.
The formula to find the area of a circle is (While π symbolizes the exact value of pi and r symbolizes radius.) πr2
NOTES:
π = c/d (c = circumference [Perimeter of a circle] d = diameter [Twice the radius, think of it as a 'height' of sorts])
So area of circle can also be portrayed as cr2/d
Now, the hard part. Setting the two formulas equal to each other.
ap/2 = cr2/d
Let's think of these two equations a little differently.
Let's replace c [circumference] with p [perimeter] Since they are basically the same thing.
Let's replace r [radius] with a [apothegm] since, once again, they are basically the same thing.
In replacing radius with apothegm, we must also say that since d [diameter] is 2r:
d [diameter] is now 2a [apothegm multiplied by two.]
Let's see what we get:
ap/2 = a2p/2a
In the second equation [a2p/2a], you'll notice that we may now cancel the 'a' that is within the denominator of the fraction and one of the 'a's within the numerator of the fraction:
a21p/2a
Thus, we find that they are equal. Such that:
ap/2 = ap/2
Does this provide concrete evidence that a circle is, in fact, a polygon?
