What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane?
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I assume you're talking about spinning the pizza around the axis a pizza-dough spinner would spin it. The moment of inertia of a solid disk is $\frac{mr^2}{2}$. If it spins, each slice contributes angular momentum proportional to $\theta$. The slice's mass is also proportional to $\theta$ so the moment of inertia of a slice of pizza about its tip is also $\frac{mr^2}{2}$ where $m$ is now the mass of the slice. We can use the parallel axis theorem to find the moment of inertia through the slice's center of mass. The center of mass is displaced a distance $\frac{2}{3}r\mathrm{sinc}\frac{\theta}{2}$ from the tip, so the moment of inertia through this axis is $$I = \left(\frac{1}{2} + (\frac{2}{3}\mathrm{sinc}\frac{\theta}{2})^2\right)mr^2$$ The height is unimportant about this axis. The location of the center of mass of the slice of pizza comes from a blog post I wrote a while ago that uses it to prove the identity
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