How to derive the formula for moment of inertia of polygon?

AI Thread Summary
The discussion centers on deriving the formula for the moment of inertia of a polygon about its centroid, specifically referencing a formula provided by user @aridno. A dead link from @chris23 was mentioned, which previously contained a derivation of this formula. A suggestion was made to use Green's Theorem to derive the area and moments of a closed polygon by connecting points with straight-line segments. The user ultimately expressed gratitude for the hint and confirmed they successfully worked out the derivation. This highlights the collaborative nature of problem-solving in physics forums.
trytodoit
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Sorry to bring this question up again.

@aridno provides a nice formula of the moment of inertia I about the centroid in https://www.physicsforums.com/threads/calculating-polygon-inertia.25293/ as:

$$
I=\sum_{n=1}^{N}\frac{\rho}{12}||\vec{P}_{n+1}\times\vec{P}_{n}||(\vec{P}_{n+1}^{2}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})
$$

in https://www.physicsforums.com/threads/moment-of-inertia-of-a-polygon.43071/ , @chris23 provides a link for the derivation, but the link is dead.

So, can someone give me a hint on how to derive this equation?
 
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trytodoit said:
Sorry to bring this question up again.

@aridno provides a nice formula of the moment of inertia I about the centroid in https://www.physicsforums.com/threads/calculating-polygon-inertia.25293/ as:

$$
I=\sum_{n=1}^{N}\frac{\rho}{12}||\vec{P}_{n+1}\times\vec{P}_{n}||(\vec{P}_{n+1}^{2}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})
$$

in https://www.physicsforums.com/threads/moment-of-inertia-of-a-polygon.43071/ , @chris23 provides a link for the derivation, but the link is dead.

So, can someone give me a hint on how to derive this equation?

You can apply Green's Theorem in the plane to derive the regular formulas for calculating the area and first and second moments of area for the general closed polygon.
You assume that the polygon is described by a set of points connected with straight-line segments and go from there, using the definitions of area and the moments.

http://en.wikipedia.org/wiki/Polygon [for calculating area and centroids]

http://en.wikipedia.org/wiki/Second_moment_of_area
 
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@https://www.physicsforums.com/threads/how-to-derive-the-formula-for-moment-of-inertia-of-polygon.809203/members/steamking.301881/ Thanks for your hint! I worked it out.
 
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