Theorem of Parallel Axes and Theorem of Perpendicular Axes

The standard moment-of-inertia expressions collected in Table 3 (as derived, for example, for the ring and disc in sections 1.5.1-1.5.2) are all about an object's own natural AXIS OF SYMMETRY. But in practice we very often need the moment of inertia about some OTHER axis -- for instance, a rod rotated about one of its ENDS rather than about its centre (an axis PARALLEL to, but offset from, the axis of symmetry), or a disc or ring rotated about one of its DIAMETERS rather than about its own central axis perpendicular to its plane (an axis PERPENDICULAR to the original axis of symmetry, rather than merely offset from it). Two theorems, both provable directly from the defining integral $I=\int r^2\,dm$ using nothing more than geometry, handle exactly these two kinds of transformation: the theorem of parallel axes (for any rigid object, relating I about any axis to I about a PARALLEL axis through the centre of mass) and the theo …