A = ¶_x r²

    Using the above formula, inserting the appropriate poly pi values from the previous chapter, and having radii as, r = 5 for Fig-5...  I would get the following:

Triangle	A = 129.9
Square	A = 100.0
Pentagon	A = 90.8
Hexagon	A = 86.6
Heptagon	A = 84.3
Octagon	A = 82.8
Nonagon	A = 81.9
Decagon	A = 81.2
Circle	A = 78.5

    To round out the calculations for the above Polygons, Fig-5, we need to find the Circumference and One Side length...

C = ¶_x * 2r

&

S = C / n_s

    Thus we hold true to a common formula of circumference for polygon and circle, with 2r being the diameter of both.  "S" equals one polygon side, and n_s equals the number of sides for the appropriate polygon.  The values for above are:

Triangle	C = 51.96	S = 17.32
Square	C = 40.	S = 10.
Pentagon	C = 36.33	S = 7.27
Hexagon	C = 34.64	S = 5.77
Heptagon	C = 33.7	S = 4.8
Octagon	C = 33.1	S = 4.1
Nonagon	C = 32.76	S = 3.64
Decagon	C = 32.49	S = 3.25
Circle	C = 31.42	S = Infinitesimal

 * * * *

    Now that we have found the correct radius and diameter that will work for all regular polygons, and the circle (at least to find their area with one formula)___ we need to proceed to the second test formula for area of a circle.   The circle area is also found by multiplying radius times circumference and dividing by two...  A = r * C / 2.  For example: A square with a circumference of eight has a poly radius of one.  Thus one times eight divided by two equal the area of four.

    This method of finding area is usually depicted by dividing a circle into many tiny pie shaped triangles with two sides being radii, as Fig-6.  The third base side is actually an arc, but when the triangles are so small__ when the circle is unrolled with the arcs abutting each other on a horizontal line, the length of the circumference, it appears as a straight line.  The height of all the pie slices is real close to the radii.  Thus the radius times the circumference forms a rectangle area that is twice the area of the circle.  Finally dividing by two gives the circle area.

A = r * C /2

    When regular polygons are sliced into pie section triangles from each corner to the polygon center along the Co-radii, and then opened to have all the circumference segments end to end in a horizontal line, it works out great.  When multiplied, the height of the rectangle is the poly radius, and the circumference is the width of the rectangle, and dividing by two gives the exact area.  Fig-7

    This formula gives an answer to solve for circle area as if the circle did have tiny flat sides, instead of the arcs mentioned, as the polygon... ?  Hum?  You try the math... radius times circumference length... divided by two... should give the same area as their respective same poly pi values.

* * * *

    At this point I am going to jump ahead a bit.  I actually do not believe now, or ever, that the circle has tiny flat sides.  I am telling you this because you probably don't believe it either!  However, the math I am doing herein leads to explanations of actions performed by circles (wheels) that are often not well known, or explained.  I will lead you to a seemingly paradoxical situation, and then we can muse about the reality of our Universe and the fantasy of perfect math.