We study the effective elastic moduli of two-dimensional composite materials containing polygonal holes. In the analysis we use a complex variable method of elasticity involving a conformal transformation. Then we take a far field result and derive the effective elastic constants of composites with a dilute concentration of polygonal holes. In the discussion we use the recently-stated Cherkaev-Lurie-Milton theorem, which gives general relations between the effective elastic constants of two-dimensional composites. We also discuss known results for elliptical holes in the context of the present work.