I have calculated the moment of inertia about the origin of a polygon based on its vertices.
The equation is
1/12 * sum { (y_{i+1} - y_{i} )(x_{i+1} + x_{i})(x_{i+1}^2 + x_{i}^2)
- (x_{i+1} - x_{i} )(y_{i+1} + y_{i})(y_{i+1}^2 + y_{i}^2)
I put the details of its derivation on my web...