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We consider ensembles of planar maps with two marked vertices at distance kfrom each other and look at the closed line separating these vertices and lyingat distance d from the first one (d<k). This line divides the map into twocomponents, the hull at distance d which corresponds to the part of the maplying on the same side as the first vertex and its complementary. The number offaces within the hull is called the hull volume and the length of theseparating line the hull perimeter. We study the statistics of the hull volumeand perimeter for arbitrary d and k in the limit of infinitely large planarquadrangulations, triangulations and Eulerian triangulations. We consider moreprecisely situations where both d and k become large with the ratio d/kremaining finite. For infinitely large maps, two regimes may be encountered:either the hull has a finite volume and its complementary is infinitely large,or the hull itself has an infinite volume and its complementary is of finitesize. We compute the probability for the map to be in either regime as afunction of d/k as well as a number of universal statistical laws for the hullperimeter and volume when maps are conditioned to be in one regime or theother.

Refined universal laws for hull volumes and perimeters in large planar maps