  We present bolometric fluxes and angular diameters for over 1.6 million stars
in the {\it Tycho-2} catalog, determined using previously-determined empirical
color-temperature and color-flux relations. We vet these relations via full
fits to the full broadband spectral energy distributions for a subset of
benchmark stars, and perform quality checks against the large set of stars for
which spectroscopically-determined parameters are available from LAMOST, RAVE,
and/or APOGEE. We then estimate radii for the 355,502 {\it Tycho-2} stars in
our sample whose \emph{Gaia} DR1 parallaxes are precise to $\lesssim$10\%. For
the 64,960 of these stars with external spectroscopic information, we achieve
median uncertainties on the effective temperatures, bolometric fluxes, angular
diameters, and radii of $1.0\%$, $1.4\%$, $2.4\%$, and $7.5\%$, respectively.
For the 290,542 remaining stars, we achieve median uncertainties of $0.9\%$,
$1.3\%$, $2.2\%$, and $7.5\%$, respectively. These stellar parameters are shown
to be reliable for stars with $T_{\rm eff}\lesssim$7000~K. The over half a
million bolometric fluxes and angular diameters presented here will serve as an
immediate trove of empirical stellar radii with the {\it Gaia\/} second data
release, at which point effective temperature uncertainties will dominate the
radius uncertainties. Already, dwarf, subgiant, and giant populations are
readily identifiable in our purely empirical luminosity-effective temperature
(theoretical) Hertzsprung-Russell diagrams.
