Areal Optimization of Polygons

We will first solve the following problem analytically: given a piece of wire of specified length, we will find where the wire should be cut and bent to form two regular polygons not necessarily having the same number of sides, so that the combined area of the polygons thus formed is maximized, minimized, greater than, and less than a specified area. We will extend the results to the cases where the wire is divided into three and finally into an arbitrary number of segments. The second problem we will solve is as follows: two wires of specified length are to be bent into two regular polygons whose total number of sides is fixed. We will determine how the total number of polygonal sides are to be allocated between the wires so that the total area of the polygons is maximized. We will extend the results found here to the case where we are given any number of wires of specified length.