  Using Density Functional Theory we theoretically study the orientational
properties of uniform phases of hard kites -- two isosceles triangles joined by
their common base. Two approximations are used: Scaled Particle Theory, and a
new approach which better approximates third virial coefficients of
two-dimensional hard particles. By varying some of their geometrical parameters
kites can be transformed into squares, rhombuses, triangles, and also very
elongated particles, even reaching the hard-needle limit. Thus a fluid of hard
kites, depending on the particle shape, can stabilize isotropic, nematic,
tetratic and triatic phases. Different phase diagrams are calculated, including
those of rhombuses, and kites with two of their equal interior angles fixed to
$90^{\circ}$, $60^{\circ}$ and $75^{\circ}$. Kites with one of their unequal
angles fixed to $72^{\circ}$, which have been recently studied via Monte Carlo
simulations, are also considered. We find that rhombuses and kites with two
equal right angles and not too large anisometry stabilise the tetratic phase
but the latter stabilize it to a much higher degree. By contrast, kites with
two equal interior angles fixed to $60^{\circ}$ stabilize the triatic phase to
some extent, although it is very sensitive to changes in particle geometry.
Kites with the two equal interior angles fixed to $75^{\circ}$ have a phase
diagram with both tetratic and triatic phases, but we show the nonexistence of
a particle shape for which both phases are stable at different densities.
Finally the success of the new theory in the description of orientational order
in kites is shown by comparing with Monte Carlo simulations for the case where
one of the unequal angles is fixed to $72^{\circ}$. These particles also
present phase diagrams with stable tetratic and triatic phases.
