Average distance from any point within a square to its diagonal Given a square with a side length "s". 
1) What is the average distance from any random point x,y within the square to the diagonal of the square?
2) if a circle with radius "r" is located at the center of the square, what is the average distance from any random point within the square to the circumference of the circle?

Thank you very much
For 1)

I'm not great at calculus but I have observed:
The average will be the same if you used a right angled triangle not an entire square.
And for a right angled triangle, the distance from the diagonal is equivalent when x+y is constant which I think really simplifies the problem.

The formula I derived was distance = (s - (x + y)) ÷ √2

Sorry I don't know how to insert math formulae.
This is the result I get from a reasonable double integral... I needed the general answer for a rectangle, which is below, where:
$w$ = rectangle width
$m$ = diagonal slope = (rectangle height / rectangle width)
For a square, $m = 1$. Hope this helps. If I get ambitious later I'll see if I can answer part two of this about the distance to the circumference of an inscribed circle.
$$ \frac13 \frac m{\sqrt{m^2+1}} w $$