# 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)

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It's good style to tell people that you're cross-posting (mathoverflow.net/questions/73556/…); else efforts will be unnecessarily duplicated. – joriki Aug 24 '11 at 15:22
Thanks for pointing out that. I've fix my post. – juhl Aug 24 '11 at 15:58

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.

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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.

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