A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Fleming
Date: 2015 Mar 20, 17:05 -0700
By diameter, I take it you mean the arc on the surface of the earth that bisects a circular patch of the surface of the earth. Use the cone angle a, at the center of the earth, for the circular patch. Then diameter is 2 · a · Re. and the circles diameter is 2 · pi · Re · sin(a).
Ratio of diameter to circumference is 2·pi·Re·sin(a) / 2·Re·a = pi·sin(a) / a .
at max diameter a = pi/2 ; cir/dia = 2
as a -> 0 cir/dia -> pi.