  The typical cell of a Voronoi tessellation generated by $n+1$ uniformly
distributed random points on the $d$-dimensional unit sphere $\mathbb S^d$ is
studied. Its $f$-vector is identified in distribution with the $f$-vector of a
beta' polytope generated by $n$ random points in $\mathbb R^d$. Explicit
formulae for the expected $f$-vector are provided for any $d$ and the
low-dimensional cases $d\in\{2,3,4\}$ are studied separately. This implies an
explicit formula for the total number of $k$-dimensional faces in the spherical
Voronoi tessellation as well.
