In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.
