  We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth
Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of
least areas circumscribed by simple closed curves $\gamma \subset \partial M$
uniquely determines the metric $g$, under some additional geometric
assumptions. These are that $g$ is either a) $C^3$-close to Euclidean or b)
satisfies much weaker geometric conditions which hold when the manifold is to a
sufficient degree either thin, or straight. %sufficiently thin.
  In fact, the least area data that we require is for a much more restricted
class of curves $\gamma\subset \partial M$. We also prove a corresponding local
result: assuming only that $(M,g)$ has strictly mean convex boundary at a point
$p\in\partial M$, we prove that knowledge of the least areas circumscribed by
any simple closed curve $\gamma$ in a neighbourhood $U\subset \partial M$ of
$p$ uniquely determines the metric near $p$. Additionally, we sketch the proof
of a global result with no thin/straight or curvature condition, but assuming
the metric admits minimal foliations "from all directions".
  The proofs rely on finding the metric along a continuous sweep-out of $M$ by
area-minimizing surfaces; they bring together ideas from the 2D-Calder\'on
inverse problem, minimal surface theory, and the careful analysis of a system
of pseudo-differential equations.
