  We study the inflated phase of two dimensional lattice polygons with fixed
perimeter $N$ and variable area, associating a weight $\exp[pA - Jb ]$ to a
polygon with area $A$ and $b$ bends. For convex and column-convex polygons, we
show that $<A >/A_{max} = 1 - K(J)/\tilde{p}^2 +
\mathcal{O}(\rho^{-\tilde{p}})$, where $\tilde{p}=pN \gg 1$, and $\rho<1$. The
constant $K(J)$ is found to be the same for both types of polygons. We argue
that self-avoiding polygons should exhibit the same asymptotic behavior. For
self-avoiding polygons, our predictions are in good agreement with exact
enumeration data for J=0 and Monte Carlo simulations for $J \neq 0$. We also
study polygons where self-intersections are allowed, verifying numerically that
the asymptotic behavior described above continues to hold.
