Calculating moment of inertia in 2d planar polygon I've derived equations for 2d polygon's moment of inertia using Green's Theorem (constant density \rho)
$$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$
$$I_x = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) ( x_{i+1} y_i - x_i y_{i+1} )$$
And I'm trying to add them up for calculating $I_0 = I_x + I_y$.
$$I_0 = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 - y_i^2 + x_i x_{i+1} - y_i y_{i+1} + x_{i+1}^2 - y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$
But I found different(?) equation for $I_0$ on the internet. and many people says below equation is correct.
$$I_0 = \frac{\rho}{6} \frac{ \sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i ) }{ \sum_{i=0}^{i=N-1} ( x_i y_{i+1} - x_{i+1} y_i ) }$$
So I'm confusing now. I think my equations for $I_x$ and $I_y$ is correct.
But how am I gonna calculate for $I_0$ (moment of inertia with respect to origin axis). 
I couldn't prove both equations are equal.
Could you help me out please ?
(This post has been cross-posted at https://math.stackexchange.com/questions/59470/calculating-moment-of-inertia-in-2d-planar-polygon)
 A: Sorry for my mistake. both equations were slightly incorrect.
Let me write the correct equations
$$I_y = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$
$$I_x = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$
$$I_0 = \frac{\rho}{12}\sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i )$$
and 
$$I_0 = \frac{m}{6} \frac{ \sum_{i=0}^{i=N-1} ( x_i^2 + y_i^2 + x_i x_{i+1} + y_i y_{i+1} + x_{i+1}^2 + y_{i+1}^2 ) ( x_i y_{i+1} - x_{i+1} y_i ) }{ \sum_{i=0}^{i=N-1} ( x_i y_{i+1} - x_{i+1} y_i ) }$$
Note that the latter equation changed the mass density term($\rho$) to mass(m). 
Both equations are equal.
A: Hi guys, it might be worth mentioning w.r.t. which axis this is expressed and the direction of the axis. Let me also share this link from wikipedia that reports a similar expression: http://en.wikipedia.org/wiki/List_of_moments_of_inertia
