In this paper we solve the following optimization problem: Given a simple polygon *P*, what is the maximum-area polygon that is axially symmetric and is contained by *P*? We propose an algorithm for solving this problem, analyze its complexity, and describe our implementation of it (for the case of a convex polygon). The algorithm is based on building and investigating a planar map, each cell of which corresponds to a different configuration of the inscribed polygon. We prove that the complexity of the map is *O(n ^{4})*, where

*n*is the complexity of

*P*. For a convex polygon the complexity is

*O(n*in the worst case.

^{3})