Here is a general way to think about these kind of problems.  The Minkowski theorem says that a polytope is uniquely determined by its normals and volumes of the facets.  You can loose some of these conditions and ask for the optimum isoperimetric ratio.  In this case, in $\Bbb R^2$ you forget normals and conclude that inscribed polygon with given side lengths is optimal.  A classical Lindelöf theorem does the opposite: in $\Bbb R^d$, it says that the optimal polytope with prescribed normals are circumscribed around the sphere (see e.g. <a href="http://www.springerlink.com/content/m388x8344gtx3731/">here</a>, Section 18.3).