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For a ring torus of major radius $A>0$, circular torus coordinates use distance $r$ and angle $\alpha$ in the circular cross-section, and azimuth $\beta$ around the central axis. The coordinate tangent vectors are the three partial derivatives of the displayed position. They are mutually orthogonal, with scale factors of orthogonal coordinates $1,r,A+r\sin \alpha$. For $0<r<A$ their ordered Jacobian determinant is $r(A+r\sin \alpha )>0$, so $dV=r(A+r\sin \alpha )drd\alpha d\beta$. At $r=0$ the cross-section angle degenerates. These are tubular coordinates for a circular torus, rather than the confocal coordinate convention also called toroidal coordinates.

The coordinate tangent vectors are the partial derivatives of the position vector:Their pairwise dot products vanish. Their lengths, the scale factors of orthogonal coordinates, are $1,r,1+r\sin \alpha$. Since $R<1$, the last is positive throughout the body. Thus these circular torus coordinates are orthogonal wherever $r>0$; at $r=0$ the cross-section angle degenerates, as in ordinary polar coordinates.

The ordered Jacobian determinant isIts sign is positive for $r>0$, so the volume element is $dV=r(1+r\sin \alpha )drd\alpha d\beta$. The coordinate degeneracy on the central circle has zero volume and does not obstruct volume integration.

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.