  Recent stellar occultations have allowed accurate instantaneous size and
apparent shape determinations of the large Kuiper belt object (50000)~Quaoar
and the detection of two rings with spatially variable optical depths. In this
paper we present new visible range light curve data of Quaoar from the
Kepler/K2 mission, and thermal light curves at 100 and 160 $\mu$m obtained with
Herschel/PACS. The K2 data provide a single-peaked period of 8.88 h, very close
to the previously determined 8.84 h, and it favours an asymmetric double-peaked
light curve with a 17.76 h period. We clearly detected a thermal light curve
with relative amplitudes of $\sim$10% at 100 and at 160 $\mu$m. A detailed
thermophysical modelling of the system shows that the measurements can be best
fit with a triaxial ellipsoid shape, a volume-equivalent diameter of 1090 km,
and axis ratios of a/b = 1.19 and b/c = 1.16. This shape matches the published
occultation shape}, as well as visual and thermal light curve data. The
radiometric size uncertainty remains relatively large ($\pm$40 km) as the ring
and satellite contributions to the system-integrated flux densities are
unknown. In the less likely case of negligible ring or satellite contributions,
Quaoar would have a size above 1100 km and a thermal inertia $\leq$ 10
Jm$^{-2}$K$^{-1}$s$^{-1/2}$. A large and dark Weywot in combination with a
possible ring contribution would lead to a size below 1080\,km in combination
with a thermal inertia $\gtrsim$ 10 Jm$^{-2}$K$^{-1}$s$^{-1/2}$, notably higher
than that of smaller Kuiper belt objects with similar albedo and colours. We
find that Quaoar's density is in the range 1.67-1.77 g/cm$^3$, significantly
lower than previous estimates. This density value closely matches the
relationship observed between the size and density of the largest Kuiper belt
objects.
