  In this paper we quantify the temporal variability and image morphology of
the horizon-scale emission from Sgr A*, as observed by the EHT in 2017 April at
a wavelength of 1.3 mm. We find that the Sgr A* data exhibit variability that
exceeds what can be explained by the uncertainties in the data or by the
effects of interstellar scattering. The magnitude of this variability can be a
substantial fraction of the correlated flux density, reaching $\sim$100\% on
some baselines. Through an exploration of simple geometric source models, we
demonstrate that ring-like morphologies provide better fits to the Sgr A* data
than do other morphologies with comparable complexity. We develop two
strategies for fitting static geometric ring models to the time-variable Sgr A*
data; one strategy fits models to short segments of data over which the source
is static and averages these independent fits, while the other fits models to
the full dataset using a parametric model for the structural variability power
spectrum around the average source structure. Both geometric modeling and
image-domain feature extraction techniques determine the ring diameter to be
$51.8 \pm 2.3$ $\mu$as (68\% credible intervals), with the ring thickness
constrained to have an FWHM between $\sim$30\% and 50\% of the ring diameter.
To bring the diameter measurements to a common physical scale, we calibrate
them using synthetic data generated from GRMHD simulations. This calibration
constrains the angular size of the gravitational radius to be
$4.8_{-0.7}^{+1.4}$ \mathrm{\mu as}, which we combine with an independent
distance measurement from maser parallaxes to determine the mass of Sgr A* to
be $4.0_{-0.6}^{+1.1} \times 10^6$ M$_{\odot}$.
