  The plethora of recent cosmologically relevant data has indicated that our
universe is very well fit by a standard Friedmann-Lema\^{i}tre-Robertson-Walker
(FLRW) model, with $\Omega_{M} \approx 0.27$ and $\Omega_{\Lambda} \approx
0.73$ -- or, more generally, by nearly flat FLRW models with parameters close
to these values. Additional independent cosmological information, particularly
the maximum of the angular-diameter (observer-area) distance and the redshift
at which it occurs, would improve and confirm these results, once sufficient
precise Supernovae Ia data in the range $1.5 < z < 1.8$ become available. We
obtain characteristic FLRW closed functional forms for $C = C(z)$ and
$\hat{M}_0 = \hat{M}_0(z)$, the angular-diameter distance and the density per
source counted, respectively, when $\Lambda \neq 0$, analogous to those we have
for $\Lambda = 0$. More importantly, we verify that for flat FLRW models
$z_{max}$ -- as is already known but rarely recognized -- the redshift of
$C_{max}$, the maximum of the angular-diameter-distance, uniquely gives
$\Omega_{\Lambda}$, the amount of vacuum energy in the universe, independently
of $H_0$, the Hubble parameter. For non-flat models determination of both
$z_{max}$ and $C_{max}$ gives both $\Omega_{\Lambda}$ and $\Omega_M$, the
amount of matter in the universe, as long as we know $H_0$ independently.
Finally, determination of $C_{max}$ automatically gives a very simple
observational criterion for whether or not the universe is flat -- presuming
that it is FLRW.
