The expected distance $d$ of randomly selected points within a unit square to the square's center is

$d = \frac{1}{6} P$

where P is the universal parabolic constant

$P = \sqrt{2} + \ln{\left(1+\sqrt{2}\right)} = 2.2955871 \dots $

see

- http://mathworld.wolfram.com/SquarePointPicking.html
- http://mathworld.wolfram.com/UniversalParabolicConstant.html

Is this a mere coincidence or is there an (intuitive) reason why this constant shows up in the solution to this problem?