A maximal planar graph is a graph which can be embedded in the plane such that every face of the graph is a triangle. The center of a graph is the subgraph induced by the vertices of minimum eccentricity. We introduce the notion of quasi-eccentric vertices, and use this to characterize maximal planar graphs that are the center of some planar graph. We also present some easier to check only necessary / only sufficient conditions for planar and maximal planar graphs to be the center of a planar graph. Finally, we use the aforementioned characterization to prove that all maximal planar graphs of order at most 8 are the center of some planar graph -- and this bound is sharp.