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 $\displaystyle r$?

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

. . Let $\displaystyle B$ be the South Pole.

Let the other point be $\displaystyle P.$

Consider the great circle through $\displaystyle A$ and $\displaystyle P.$

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 $\displaystyle P$ can be anywhere on the arc $\displaystyle \overline{APB}.$

Since half the points are above the Equator $\displaystyle OE$ and half are below,

. . the average distance would be: .$\displaystyle \text{arc}\overline{AE}$

Therefore, average distance .$\displaystyle = \;\frac{1}{4}(2\pi r) \;=\;\frac{1}{2}\pi r$

. . just as Haytham predicted . . .