Answer:
The proof that πk(C1)=πk(C2) of course would just apply the similarity of polygons and the behavior of length and area for changes of scale. This argument does not assume a limit-based theory of length and area, because the theory of length and area for polygons in Euclidean geometry only requires dissections and rigid motions ("cut-and-paste equivalence" or equidecomposability). Any polygonal arc or region can be standardized to an interval or square by a finite number of (area and length preserving) cut-and-paste dissections. Numerical calculations involving the πk, such as ratios of particular lengths or areas, can be understood either as applying to equidecomposability classes of polygons, or the standardizations. In both interpretations, due to the similitude, the results will be the same for C1 and C2.
