  We develop a method to compute the generating function of the number of
vertices inside certain regions of the Uniform Infinite Planar Triangulation
(UIPT). The computations are mostly combinatorial in flavor and the main tool
is the decomposition of the UIPT into layers, called the skeleton
decomposition, introduced by Krikun. In particular, we get explicit formulas
for the generating functions of the number of vertices inside hulls (or
completed metric balls) centered around the root, and the number of vertices
inside geodesic slices of these hulls. We also recover known results about the
scaling limit of the volume of hulls previously obtained by Curien and Le Gall
by studying the peeling process of the UIPT.
