Let's consider the inscribed square of the circle such that its vertices describe the chords to its 4 quadrants and where area of the circle is, say, 64 units. Area of the inscribed square is given as 64/0.5 pi = 40.5 square units and where each line of the square defines the chord length to a quadrant of the circle. That chord length is given by the trianglature formulae of r*sqrt 2 = chord length; chord length*pi/4*sqrt 2 = quadrant arc length.

 

The trianglature formulae derives from aeronautics engineer and writer E.P. LeRoy.

I keep working on it and have now come up with a formulae giving further evidence that it is not pi that officiates in determining dimension of the circle but root 2. Here is the formulae I come up with:

 

Sqrt of area/sqrt pi = radius; r^2*pi = area. Thus for the given area of 64:

 

8/sqrt pi = 4.5135....radius; r^2*pi = 64 area.

 

Root 2 Rules!

I keep working on it and have now come up with a formulae giving further evidence that it is not pi that officiates in determining dimension of the circle but root 2. Here is the formulae I come up with:

 

Sqrt of area/sqrt pi = radius; r^2*pi = area. Thus for the given area of 64:

 

8/sqrt pi = 4.5135....radius; r^2*pi = 64 area.

 

Root 2 Rules!

Aye Caramba!... Hold on to your hat everyone! It would appear that not only dioes root 2 rule - but that it is as well sacosanct! - for it is now seen that applying the formulae to any area and applying any known pi value whatsoever results in deriving the same area. Root 2 would appear indeed to be sacrosanct! - pi but the ratio of line to arc. What say y'all?