  The Newton--Okounkov body of a big divisor D on a smooth surface is a
numerical invariant in the form of a convex polygon. We study the geometric
significance of the shape of Newton--Okounkov polygons of ample divisors,
showing that they share several important properties of Newton polygons on
toric surfaces. In concrete terms, sides of the polygon are associated to some
particular irreducible curves, and their lengths are determined by the
intersection numbers of these curves with D.
  As a consequence of our description we determine the numbers k such that D
admits some k-gon as a Newton--Okounkov body, elucidating the relationship of
these numbers with the Picard number of the surface, which was first hinted at
by work of K\"uronya, Lozovanu and Maclean.
