Tie a weight to one end of the string.
Pinch the unweighted end of the string between two fingers against one point on the perimeter of the polygon. Let the polygon and the string hang loosely. Mark a line along the string in pencil.
Repeat Step 2 using a different point on the perimeter of the polygon, marking a second line along the string. For best results, pick a point that is not very close to either end of the first line that you drew. The point where the two lines cross is the center of gravity.
Remove the weight from the string. Place a small drop of glue at the point where the two lines cross on the polygon. Carefully attach one tip of the string to the glue and let it dry. When you lift the string, the polygon should balance on the end of it.