Let $θ= [0; a_1, a_2, \dots]$ be the continued fraction expansion of an irrational real number $θ\in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $θ$ is the limit of a sequence of finite words $(M_k)_{k \ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $θ$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b \ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $ξ$ whose $b$-ary expansion is a Sturmian word ${\bf s}$ over the alphabet $\{0, b-1\}$. This extends a classical result of Böhmer who considered only the case where ${\bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $ξ$ in terms of the slope and the intercept of ${\bf s}$.