  The black hole area theorem implies that when two black holes merge, the area
of the final black hole should be greater than the sum of the areas of the two
original black holes. We examine how this prediction can be tested with
gravitational-wave observations of binary black holes. By separately fitting
the early inspiral and final ringdown stages, we calculate the posterior
distributions for the masses and spins of the two initial and the final black
holes. This yields posterior distributions for the change in the area and thus
a statistical test of the validity of the area increase law. We illustrate this
method with a GW150914-like binary black hole waveform calculated using
numerical relativity, and detector sensitivities representative of both the
first observing run and the design configuration of Advanced LIGO. We obtain a
$\sim74.6\%$ probability that the simulated signal is consistent with the area
theorem with current sensitivity, improving to $\sim99.9\%$ when Advanced LIGO
reaches design sensitivity. An important ingredient in our test is a method of
estimating when the post-merger signal is well-fit by a damped sinusoid
ringdown waveform.
