How valuable is range

After an afternoon of trying to calculate exactly how often two randomly placed 'Mechs would be any given distance from each other, I noticed that the distribution was going to look something like sqrt(x^2 + y^2 - r^2), where x and y are the length and width of a rectangular board, and r is the distance between any two randomly selected points on that board. (The actual equation may feature arcsins, and would have some sort of scaling factor, but I figure this is close enough to fudge the rest.)

Pretty sure it only works this way because the maps have square borders... there are just a ton of ways that you can place two 'Mechs at range 1, and only a little fewer at range 2, but as your range gets longer you start getting trapped going corner-to-corner. There's not many ways to do that, so the curve drops off sharply. It zeroes out when you finally go off the map.

A circular border would have a different distribution (sqrt(xy)-r, maybe?) and the curve for an infinite map might instead march continuously upward.

Of course, this is just the distance between two randomly selected points. In a boardgame, those points won't be random, and the terrain is rarely so perfectly flat and open. For BattleTech, when the scenario prevents kiting, I'd guess that range decreases in usefulness more or less linearly from range 1 and zeroes around range 30. (You could calculate these values exactly, given a known distribution of terrain and combatants and victory conditions, but it'd be hassle because those things change so often.)