For a line of latitude
the distance around the line of latitude is
pRsinq and the distance over the pole is 2Rq.
Comparing these distances:
for all
q except at the equator when q = p/2 when the
distances are equal. This shows that the great circle distance is
always shorter as of course the equator and the lines of longitude
are all great circles.
A graph of the ratio of these distances shows that
tends to a limit as q® 0.
As we know
as q® 0 we
see that this limit is 2/p which is 0.6366 to 4 sig figs.