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Calculate the moment of inertia for a given convex two-dimensional polygon

I have been dealing with this issue for hours and for some reason I could not find a solution.

For a convex polygon that is defined as an array of points clockwise around the polygon's centroid, how can I calculate the polygon's moment of inertia?

I was able to find an equation for various shapes like rectangles or circles, but not for an arbitrary convex polygon.

For example, the moment of inertia of a rectangle rotating around its center of gravity with mass m, height h and width w is calculated as:

Moment of Inertia for a Rectangle

I'm looking for a similar function fomula / algorithm, but instead of this convex polygon.

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The equation for moment of inertia is pretty simple and you can find an explanation here: https://en.wikipedia.org/wiki/Moment_of_inertia

It was used, for example, to get the equation you quoted.

As Andy Newman mentioned, a convex polygon can be thought of as consisting of triangles. But summing their individual inertia is not a solution - it will not give the correct equation for the axis of rotation.



So what you basically need to do is get the equation for your polygon about the axis of rotation that you want to rotate it to.

You may find one of the following theorems helpful, depending on the type of form you have in mind:

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