  We conduct a series of measurements on the thermodynamic properties of an
optically-trapped strongly interacting Fermi gas, including the energy $E$,
entropy $S$, and sound velocity $c$. Our model-independent measurements of $E$
and $S$ enable a precision study of the finite temperature thermodynamics. The
$E(S)$ data are directly compared to several recent predictions. The
temperature in both the superfluid and normal fluid regime is obtained from the
fundamental thermodynamic relation $T=\partial E/\partial S$ by parameterizing
the $E(S)$ data. Our $E(S)$ data are also used to experimentally calibrate the
endpoint temperatures obtained for adiabatic sweeps of the magnetic field
between the ideal and strongly interacting regimes. This enables the first
experimental calibration of the temperature scale used in experiments on
fermionic pair condensation. Our calibration shows that the ideal gas
temperature measured for the onset of pair condensation corresponds closely to
the critical temperature estimated in the strongly interacting regime from the
fits to our $E(S)$ data. The results are in very good agreement with recent
predictions. Finally, using universal thermodynamic relations, we estimate the
chemical potential and heat capacity of the trapped gas from the $E(S)$ data.
