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.