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Expected Distance

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What is the expected distance of two points in a unit square?

What is the expected distance of two points in a regular unit pentagon?

Does the expected distance between two points in a regular unit n-gon converge as n goes to infinity?

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If you mean unit area, the mean distance between two interior points of a unit regular n-gon converges to that of a disk.

If you mean unit side, then the mean distance diverges, since that type of unit regular n-gon becomes the entire plane.

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If you mean unit area, the mean distance between two interior points of a unit regular n-gon converges to that of a disk.

If you mean unit side, then the mean distance diverges, since that type of unit regular n-gon becomes the entire plane.

how do we find the mean distance for a square? and for a regular unit pentagon? seems very very tough to me :(

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Empirically, by simulation. Or do some area integrals.

For an equilateral triangle with unit sides it's 0.364791843300...
For a square with unit side it's 0.521405433....
For a circle with unit radius it's 0.905414787...

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