We need to determine the moment of inertia perpendicular to the ring, passing though it's center.

First let’s take a point on the rim, this point is perpendicular to the ring and moment of Inertia will be-

If the ring passes through its particles at a distance of r, it will be parallel to the axis.

By applying parallel axis theorem

Now for a new line, radius of gyration will be a, therefore

Equating both sides of equation for I’-