Second vertical moment of a solid ring torus (${\int }_V{z}^{2}dV={\pi }^{2}A{R}^4/2$)

A solid torus of major radius $A$ and minor radius $R<A$, centred on the horizontal plane, has volume $2{\pi }^{2}A{R}^2$. In circular torus coordinates, $z=r\cos \alpha$. Integrating $z^2$ with the Jacobian determinant $r(A+r\sin \alpha )$ gives ${\int }_V{z}^{2}dV=({\pi }^2/2)A{R}^4$; the part odd in $\sin \alpha$ vanishes. Thus the mean squared height is $R^2/4$. The identity is useful for moments of inertia and flux calculations using the divergence theorem.