  We revisit the shapes of the throats of wormholes, including thin-shell
wormholes (TSWs) in $2+1-$dimensions. In particular, in the case of TSWs this
is done in a flat $2+1-$dimensional bulk spacetime by using the standard method
of cut-and-paste. Upon departing from a pure time-dependent circular shape
i.e., $r=a\left( t\right) $ for the throat, we employ a $% \theta -$dependent
closed loop of the form $r=R\left( t,\theta \right) ,$ and in terms of $R\left(
t,\theta \right) $ we find the surface energy density $\sigma $ on the throat.
For the specific convex shapes we find that the total energy which supports the
wormhole is positive and finite. In addition to that we analyze the general
wormhole's throat. By considering a specific equation of $r=R\left( \theta
\right) $ instead of $r=r_{0}=const.,$ and upon certain choices of functions
for $R\left( \theta \right) $ we find the total energy of the wormhole to be
positive.
