Diameter vs. radius
Today, the radius of a circle is widely accepted as its most important measure. For example, the “unit
circle,” around which radian angles are measured, has a radius of 1. But at the time when the greek letter pi
was first being associated with the circle’s circumference, it was thought that the diameter was the important measure of
a circle. Since the diameter is exactly twice the radius, this introduces an arbitrary factor of 2 into pirelated
phenomena, such as:
•   One time around the unit circle is
2pi. 
•   1/4 of the way around the unit circle (a right angle)
is 2pi/4. 
•   1/n of the way around the unit circle is
2pi/n. 
•   One full cycle of a sine or cosine wave spans a width
of 2pi. 
•   A half cycle of a sine or cosine wave spans a width
of pi. 
•   Radian angles of 0 and 2pi are equal; 0 and pi are not. 
•   etc. 
If early mathematicians had measured their circles by radius, they perhaps would have assigned pi the value of 6.283..,
and this factor of 2 would not be necessary.
