For a circular cross-section of radius , Eq.(B.11) tells us that the squared radius of gyration about any line passing through the center of the cross-section is given by
Using the elementrary trig identity , we readily derive
The first two terms of this expression contribute zero to the integral from 0 to , while the last term contributes , yielding
Thus, the radius of gyration about any midline of a circular cross-section of radius is
For a circular tube in which the mass of the cross-section lies within a circular annulus having inner radius and outer radius , the radius of gyration is given by