Summary: We derive the limiting distribution of the barycenter \(b_n\) of an i.i.d. sample of \(n\) random points on a planar cone with angular spread larger than \(2\pi\). There are three mutually exclusive possibilities: (i) (fully sticky case) after a finite random time the barycenter is almost surely at the origin; (ii) (partly sticky case) the limiting distribution of \(\sqrt{n} b_n\) comprises a point mass at the origin, an open sector of a Gaussian, and the projection of a Gaussian to the sector’s bounding rays; or (iii) (nonsticky case) the barycenter stays away from the origin and the renormalized fluctuations have a fully supported limit distribution-usually Gaussian but not always. We conclude with an alternative, topological definition of stickiness that generalizes readily to measures on general metric spaces.