Perpendicular axes theorem (for a planar body): the moment of inertia about an axis perpendicular to the plane equals the sum of the moments of inertia about two mutually perpendicular axes in the plane meeting at the same point: Iz = Ix + Iy. For a ring, Iz (about central axis) = M R^2, and by symmetry Ix = Iy = I(diameter). So M R^2 = 2 I(diameter), giving I(diameter) = M R^2 / 2.

Iz = Ix + Iy

The radius of gyration K is the distance from the axis at which the whole mass of the body can be assumed to be concentrated so as to give the same moment of inertia: I = M K^2, hence K = sqrt(I / M). Its SI unit is the metre.

K = sqrt(I / M)

For a ring about its central axis, I = M R^2 = 2 * (0.5)^2 = 2 * 0.25 = 0.5 kg m^2.

I = M R^2