  We study Newton polytopes of cluster variables in type A_n cluster algebras,
whose cluster and coefficient variables are indexed by the diagonals and
boundary segments of a polygon. Our main results include an explicit
description of the affine hull and facets of the Newton polytope of the Laurent
expansion of any cluster variable, with respect to any cluster. In particular,
we show that every Laurent monomial in a Laurent expansion of a type A cluster
variable corresponds to a vertex of the Newton polytope. We also describe the
face lattice of each Newton polytope via an isomorphism with the lattice of
elementary subgraphs of the associated snake graph.
