  Using \emph{in situ} measurements on a quasi two-dimensional, harmonically
trapped $^{87}$Rb gas, we infer various equations of state for the equivalent
homogeneous fluid. From the dependence of the total atom number and the central
density of our clouds with the chemical potential and temperature, we obtain
the equations of state for the pressure and the phase-space density. Then using
the approximate scale invariance of this two-dimensional system, we determine
the entropy per particle. We measure values as low as $0.06\,\kB$ in the
strongly degenerate regime, which shows that a 2D Bose gas can constitute an
efficient coolant for other quantum fluids. We also explain how to disentangle
the various contributions (kinetic, potential, interaction) to the energy of
the trapped gas using a time-of-flight method, from which we infer the
reduction of density fluctuations in a non fully coherent cloud.
