I trying to formalize the measure by which a circle covers "more" space than a square of equal area, in following sense.

Let G be a 1000 unit area in orthogonal dimensions x and y, with a corner point p, and whose interior points are all points {n,m} denoting the point n positive units from p in x, and m positive units from p in y. Let C be a circle, and S be square both with A=10, and {500,500} as their centers.

Let CD be a set of distinct randomly chosen points from the perimeter of C; SD be a set of randomly chosen points from the perimeter of S, and N be the cardinality of both. Let A(CD) and be a function that returns the average of the distance of each point in CD from {500,500}, the center of C, and likewise for A(SD).

Intuitively (to me, and please disprove formally if wrong):

as N approaches infinity A(CD)>A(SD)

If this is correct, can someone please formalize general numerical relationship between circles and squares from which it can be derived?