  Cluster algebras are rings with distinguished generators called cluster
variables, grouped into clusters. Each cluster variable can be written as a
Laurent polynomial in any cluster. In this paper, we study the Newton polytopes
of these Laurent polynomials. We focus on cluster algebras of types $A$ and
$D$, with frozen variables corresponding to the boundary segments of a polygon
and punctured polygon, respectively. For these cluster algebras, we show the
cluster variable Newton polytopes are saturated. For type $A$, we additionally
show that the cluster variable Newton polytopes have no non-vertex lattice
points. Our main tool is the snake graph expansion formula of
Musiker-Schiffler-Williams for cluster algebras from surfaces.
