  We develop a new method to estimate the area, and more generally the
intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$
that is close in the Hausdorff distance. This estimator enjoys a linear rate of
convergence as a function of the Hausdorff distance under mild regularity
conditions on $X$. Our approach combines tools from both geometric measure
theory and persistent homology, extending the noise filtering properties of
persistent homology from the realm of topology to geometry. Along the way, we
obtain a stability result for intrinsic volumes.
