Calculating moment of inertia in 2d planar polygon I've derived equations for a 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 a different(?) equation for $I_0$ on the internet. and many people say the equation given below 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 confused now. I think my equations for $I_x$ and $I_y$ are correct.
But how am I gonna calculate $I_0$ (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 MathOverflow)
 A: Your moments don't pass two straightforward tests: They should be invariant under reversal of the vertex order (instead they change sign); and they should be quadratic under scaling (instead they scale with the fourth power). The expression you quote from the net passes both tests, so there's a good chance it's correct.
A: Sorry for my mistake. both equations was slightly incorrect.
Let me write 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 latter equation changed mass density term($\rho$) to mass(m). 
Both equations are equal.
