  We introduce an area operator for the Moyal noncommutative plane. We find
that the spectrum is discrete, but, contrary to the expectation formulated by
other authors, not characterized by a "minimum-area principle". We show that an
intuitive analysis of the uncertainty relations obtained from Moyal-plane
noncommutativity is fully consistent with our results for the spectrum, and we
argue that our area operator should be generalizable to several other
noncommutative spaces. We also observe that the properties of distances and
areas in the Moyal plane expose some weaknesses in the line of reasoning
adopted in some of the heuristic analyses of the measurability of geometric
spacetime observables in the quantum-gravity realm.
