  In this paper, we are interested in the behavior of the typical
Poisson-Voronoi cell in the plane when the radius of the largest disk centered
at the nucleus and contained in the cell goes to infinity. We prove a law of
large numbers for its number of vertices and the area of the cell outside the
disk. Moreover, for the latter, we establish a central limit theorem as well as
moderate deviation type results. The proofs deeply rely on precise connections
between Poisson-Voronoi tessellations, convex hulls of Poisson samples and
germ-grain models in the unit ball. Besides, we derive analogous facts for the
Crofton cell of a stationary Poisson line process in the plane.
