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

1. Apr 18, 2015

### trytodoit

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?

2. Apr 18, 2015

### SteamKing

Staff Emeritus
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

3. Apr 22, 2015

### trytodoit

@https://www.physicsforums.com/threads/how-to-derive-the-formula-for-moment-of-inertia-of-polygon.809203/members/steamking.301881/ [Broken] Thanks for your hint! I worked it out.

Last edited by a moderator: May 7, 2017
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