There is a local metric. The value of length/2/r depends on how big the circle is:
Imagine the Earth is a sphere. You make circles centered in the north pole:
* If the circle is tiny, the Earth is almost flat and you get almost pi.
* If the circle is the equator, you have to walk 1/4 of length the circle from the pole to the equator, so the result is 4/2=2
* If the circle is so big that you walked almost to the south pole, the result is almost 0.
That makes perfect sense!
I guess my point is that Pi is only a minimum in the selected family of metrics that the article examines. There are plenty of other metrics where Pi is as small or as big as you want.
The great circle distance is a global geodesic on the sphere surface.