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Download Navigation CD Europe Version 10.1
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Moment of inertia of a convex polyhedron

Let’s say I have a perfectly convex and symmetric polyhedron. Consider the center of mass O of this polyhedron to be located at position (O, O). Let (a, b, c, d) be the vertices of this polyhedron. My question is, what is the moment of inertia of this polyhedron about the axis OX (which is parallel to one of the edges a, b, c, d) and perpendicular to OY. In other words, what is the moment of inertia of this polyhedron about axis OY.
I can easily find this by using the formula (PI*IV)/6 for a solid where I is the moment of inertia tensor and V is the orientation of a solid. But, I feel like this should be a well known formula given that polyhedra and finite volumes are ubiquitous in many problems. Also, using this formula is probably an overkill if the volume is large since there are simpler ways of finding this. Is there such a formula for this case or is there an upper bound on the moment of inertia about axis OY for this case?

A:

It can be shown that the moments of inertia of a symmetric polyhedron about axes parallel to edges or parallel to faces of the polyhedron are quite simple: Let the edge $e_i$ be the edge connecting vertices $v_i$ and $v_{i+1}$ (indices understood modulo $n$); then the moment of inertia $I_{x_i}(O)$ with $x_i$ a unit vector in the $i$-th edge about the $x_i$-axis is
$$
I_{x_i}(O) = 2 \frac{1}{n} \sum_{j=1}^n \hat{e}_i \cdot \hat{e}_j
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