  Using the stellar kinematic maps and ancillary imaging data from the Sydney
AAO Multi Integral field (SAMI) Galaxy Survey, the intrinsic shape of
kinematically-selected samples of galaxies is inferred. We implement an
efficient and optimised algorithm to fit the intrinsic shape of galaxies using
an established method to simultaneously invert the distributions of apparent
ellipticities and kinematic misalignments. The algorithm output compares
favourably with previous studies of the intrinsic shape of galaxies based on
imaging alone and our re-analysis of the ATLAS3D data. Our results indicate
that most galaxies are oblate axisymmetric. We show empirically that the
intrinsic shape of galaxies varies as a function of their rotational support as
measured by the "spin" parameter proxy Lambda_Re. In particular, low spin
systems have a higher occurrence of triaxiality, while high spin systems are
more intrinsically flattened and axisymmetric. The intrinsic shape of galaxies
is linked to their formation and merger histories. Galaxies with high spin
values have intrinsic shapes consistent with dissipational minor mergers, while
the intrinsic shape of low-spin systems is consistent with dissipationless
multi-merger assembly histories. This range in assembly histories inferred from
intrinsic shapes is broadly consistent with expectations from cosmological
simulations.
