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

## Main Question or Discussion Point

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:

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

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

SteamKing
Staff Emeritus
Homework Helper
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:

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

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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