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)