We describe an approximate generalization of a theorem of Comte-Lion-Rolin on volumes of $\mathbb{R}_{an}$-definable sets. Let $\mathfrak{R}$ be a polynomially bounded o-minimal expansion of the real field. Our result shows that the volume of an $\mathfrak{R}$-definable set is bounded from above and below by constant multiples of a $\mathfrak{R}_{\exp}$-definable function of its defining parameters. Joint with Ehud Hrushovski.