A hyperconvex disc of radius $$r$$ is a planar set with nonempty interior that is the intersection of closed circular discs of radius $$r$$ . A convex disc-polygon of radius $$r$$ is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius $$r$$ . We prove that the maximum area and perimeter of convex disc- $$n$$ -gons of radius $$r$$ contained in a hyperconvex disc of radius $$r$$ are concave functions of $$n$$ , and the minimum area and perimeter of disc- $$n$$ -gons of radius $$r$$ containing a hyperconvex disc of radius $$r$$ are convex functions of $$n$$ . We also consider hyperbolic and spherical versions of these statements.