ds_circular(d=7) pond : ds = 7 * 64/(45*Pi) ~ 3.1689

ds_quadratic(s=6) pond : ds = 6 * (5*Log[Sqrt[2]+1]+2+Sqrt[2])/15 ~ 3.1284

indeed

there exists a very small region for the pond-parameter, where the circular distance is smaller in despite of greater area !!!!

so the 'far reaching edges' of the quadratic pond have really relevance!

comments & rechenversuche (*gg*) available at this site .