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We study coarea inequalities for metric surfaces -- metric spaces that aretopological surfaces, without boundary, and which have locally finite Hausdorff2-measure $\mathcal{H}^2$. For monotone Sobolev functions $u\colon X \to\mathbb{R} $, we prove the inequality  \begin{equation*}  \int_{ \mathbb{R} }^{*}  \int_{ u^{-1}(t) }  g  \,d\mathcal{H}^{1}  \,dt  \leq  \kappa  \int_{ X }  g \rho  \,d\mathcal{H}^{2}  \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,}\end{equation*} where $\rho$ is any integrable upper gradient of $u$. If $\rho$is locally $L^2$-integrable, we obtain the sharp constant $\kappa=4/\pi$. Themonotonicity condition cannot be removed as we give an example of a metricsurface $X$ and a Lipschitz function $u \colon X \to \mathbb{R}$ for which thecoarea inequality above fails.

Coarea Inequality for Monotone Functions on Metric Surfaces