Given a finitely generated semigroup $S$ and subsemigroup $T$ of $S$ we define the notion of the boundary of $T$ in $S$ which, intuitively, describes the position of $T$ inside the left and right Cayley graphs of $S$. We prove that if $S$ is finitely generated and $T$ has a finite boundary in $S$ then $T$ is finitely generated. We also prove that if $S$ is finitely presented and $T$ has a finite boundary in $S$ then $T$ is finitely presented. Several corollaries and examples are given. We also include a corrected proof of a result from an earlier paper showing that finite presentability is inherited by subsemigroups with finite complement.