  Facets of the convex hull of $n$ independent random vectors chosen uniformly
at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular
focus is given on the height of the facets as well as the expected number of
facets as the dimension increases. Regimes for $n$ and $d$ with different
asymptotic behavior of these quantities are identified and asymptotic formulas
in each case are established. Extensions of some known results in fixed
dimension to the case where dimension tends to infinity are described.
