TITLE: Average distance between two randomly chosen points in unit square (without calculus)
QUESTION [24 upvotes]: Imagine that you choose two random points within a 1 by 1 square.  What is the average distance between those two points?  Using a random number generator, I'm getting a value of ~0.521402... can anyone explain why I'm getting this value, or what this number means? 
More importantly, is there a way to solve this without using calculus and/or large random sampling?

REPLY [12 votes]: When you say "without calculus", then presumably you are talking about just using something like basic algebra or Euclidean geometry. However, as you have seen from the up-voted posted answers and comments, the exact answer involves natural logarithm. It is basically impossible to define natural logarithm without some sort of calculus and/or real analysis. So it seems you need calculus (or its more formal version, real analysis) to come up with the formula for this answer.