  The interior hull of a lattice polygon is the convex closure of the lattice
points in the interior of the polygon. In this paper we give a concrete
description of the interior hull of a clean lattice parallelogram. A clean
parallelogram in $\mathbb{R}^2$ is a lattice parallelogram whose boundary
contains no lattice points other than its vertices. Using unimodular maps we
can identify a clean parallelogram with a parallelogram, $P_{a,n}$, whose
vertices are $(0,0), (1,0), (a,n)$ and $(a+1,n)$, with $0<a <n$ and
$\gcd(a,n)=1$. Following Stark's geometric approach to continued fractions we
show that the convergents of the continued fraction of $n/a$ (viewed as lattice
points) appear in a one-to-two correspondence with the vertices of the interior
hull of this parallelogram. Consequently, if the continued fraction of $n/a$
has many partial quotients, then the interior hull of the corresponding
parallelogram has many vertices.
  A pleasing consequence of our work is that we obtain an elementary geometric
interpretation of the sum of the partial quotients of the continued fraction of
$n/a$. Specifically, it is the difference between the area of the clean
parallelogram $P_{a,n}$ and the area of its interior hull.
