  A remarkable theorem due to Khovanskii asserts that for any finite subset $A$
of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a
polynomial for all sufficiently large $h$. Currently, neither the polynomial
nor what sufficiently large means are understood. In this paper we obtain an
effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}^d$
whose convex hull is a simplex; previously, such results were only available
for $d=1$. Our approach gives information about not just the cardinality of
$hA$, but also its structure, and we prove two effective theorems describing
$hA$ as a set: one answering a recent question posed by Granville and Shakan,
the other a Brion-type formula that provides a compact description of $hA$ for
all large $h$. As a further illustration of our approach, we derive a
completely explicit formula for $|hA|$ whenever $A \subset \mathbb{Z}^d$
consists of $d+2$ points.
