  We provide sharp lower bounds for the simplicial volume of compact
$3$-manifolds in terms of the simplicial volume of their boundaries. As an
application, we compute the simplicial volume of several classes of
$3$-manifolds, including handlebodies and products of surfaces with the
interval. Our results provide the first exact computation of the simplicial
volume of a compact manifold whose boundary has positive simplicial volume. We
also compute the minimal number of tetrahedra in a (loose) triangulation of the
product of a surface with the interval.
