This is interesting - it didn't occur to me to check it empirically. Can you think of a way to prove this hypothesis? Thanks.

Yes, I am sorry I meant to choose uniformly from the interior of circle. When $a=b$, the expression doesn’t look pretty (not to me at least) but the argument/procedure is simple enough (It’s described in Tuckwell’s Applications of Probability Theory). Anyway here it is. $f_X(x) = \frac{2x}{\pi r^2}( 2 \arccos(\frac{x}{2r}) - \frac{x}{r} \sqrt{ 1-{(\frac{x}{2r}) }^2} )$.

Yes Gerry, I want to compute average degree of nodes placed, roughly, in a grid where adjacency defined in terms of distance. Problem arises in sensor networks.