We address a geometric inverse problem: Consider a simply connected Riemannian 3-manifold (M,g) with boundary. Assume that given any closed loop \gamma on the boundary, one knows the area of the area-minimizer bounded by \gamma. Can one reconstruct the metric g from this information? We answer this in the affirmative in a very broad open class of manifolds. We will briefly discuss the relation of this problem with the question of reconstructing a metric from lengths of geodesics, and also with the Calderon problem of reconstructing a metric from the Dirichlet-to-Neumann operator for the corresponding Laplace-Beltrami operator. We also raise the analogous question for asymptotically hyperbolic manifolds, and the significance of their question in physics. Joint with T Balehowsky and A Nachman.