  A fundamental effect of fluid turbulence is turbulent mixing, which results
in the stretching and wrinkling of scalar isosurfaces. Thus, the area of
isosurfaces is of interest in understanding turbulence in general with specific
applications in, e.g., combustion and the identification of
turbulent/non-turbulent interfaces. We report measurements of isosurface areas
in 28 direct numerical simulations (DNSs) of homogeneous isotropic turbulence
with a mean scalar gradient resolved on up to $14256^3$ grid points with Taylor
Reynolds number $Re{_\lambda}$ ranging from 24 to 633 and Schmidt number $Sc$
ranging from 0.1 to 7. More precisely, we measure layers with very small but
finite thickness. The continuous equation we evaluate converges exactly to the
area in the limit of zero layer thickness. We demonstrate a method for
numerically integrating this equation that, for a test case with an analytical
solution, converges linearly towards the exact solution with decreasing layer
width. By applying the technique to DNS data and testing for convergence with
resolution of the simulations, we verify the resolution requirements for DNS
recently proposed by \citet{yeung18}. We conclude that isosurface areas scale
with the square root of the Taylor P\'eclet number $Pe_{\lambda}$ between
approximately 50 and 4429 with some departure from power law scaling evident
for $2.4 < Pe_{\lambda} < 50$. No independent effect of either $Re_{\lambda}$
or $Sc$ is observed. The excellent scaling of area with $Pe_{\lambda}^{1/2}$
occurs even though the probability density function (p.d.f.) of the scalar
gradient is very close to exponential for $Re_{\lambda}=98$ but approximately
lognormal when $Re_{\lambda}=633$.
