  The phase diagram of a system of monodispersed hard rectangles of size
$m\times m k$ on a square lattice is numerically determined for $m=2,3$ and
aspect ratio $k= 1,2,\ldots, 7$. We show the existence of a disordered phase, a
nematic phase with orientational order, a columnar phase with orientational and
partial translational order, and a solid-like phase with sublattice order, but
no orientational order. The asymptotic behavior of the phase boundaries for
large $k$ are determined using a combination of entropic arguments and a Bethe
approximation. This allows us to generalize the phase diagram to larger $m$ and
$k$, showing that for $k \geq 7 $, the system undergoes three entropy driven
phase transitions with increasing density. The nature of the different phase
transitions are established and the critical exponents for the continuous
transitions are determined using finite size scaling.
