Hello, Zamadatix!

I *think* I have an answer . . .

What is the average distance between two points

on the surface of a sphere with a radius ?

With no loss of generalization, let one point be at the North Pole of the sphere.

. . Let be the South Pole.

Let the other point be

Consider the great circle through and

It could pass through Greenwich Mean Time

. . (discovered by the Armenian explorer, Prime Meridian).

Code:

A
* o *
* * P
* o
* *
* *
* * - - - - * E
* O r *
* *
* *
* o *
* * *
B

Point can be anywhere on the arc

Since half the points are above the Equator and half are below,

. . the average distance would be: .

Therefore, average distance .

. . just as Haytham predicted . . .