Conical carbon nanofibers (CNFs) exist primarily as graphitic ribbons that fold into a cylindrical structure with the formation of a hollow core. Structural analysis aided by molecular modeling proves useful for obtaining a full picture of how the size of the central channel varies from fiber to fiber. From a geometrical perspective, conical CNFs possibly have cone tips that are nearly closed. On the other hand, their fiber wall thickness can be reduced to a minimum possible value that is determined solely by the apex angle, regardless of the outer diameter. A formula has been developed to express the number of carbon atoms present in conical CNFs in terms of measurable structural parameters. It appears that the energetically preferred fiber wall thickness increases not only with the apex angle, but also with the number of atoms in the constituent graphitic cones. The origin of the empirical observation that conical CNFs with small apex angles tend to have a large hollow core lies in the fact that in graphene sheets that are more highly curved the curvature-induced strain energy rises more rapidly as the fiber wall thickens.