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>)



Calculating moment of inertia in 2d planar polygon
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.


