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We revisit the shapes of the throats of wormholes, including thin-shellwormholes (TSWs) in $2+1-$dimensions. In particular, in the case of TSWs thisis done in a flat $2+1-$dimensional bulk spacetime by using the standard methodof cut-and-paste. Upon departing from a pure time-dependent circular shapei.e., $r=a\left( t\right) $ for the throat, we employ a $% \theta -$dependentclosed 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 thewormhole is positive and finite. In addition to that we analyze the generalwormhole's throat. By considering a specific equation of $r=R\left( \theta\right) $ instead of $r=r_{0}=const.,$ and upon certain choices of functionsfor $R\left( \theta \right) $ we find the total energy of the wormhole to bepositive.

2+1-dimensional traversable wormholes supported by positive energy