By symmetry the moment of inertia about every diameter is the same, say I_{d}. This means that, when the disc is in the xy plane with its centre at the origin,

I_{x} = I_{y}  +  I_{d}.

But I_{z} is the moment of inertia about the axis through O perpendicular to the disc and this is \frac{1}{2}Mr^{2}. Hence

\frac{1}{2}Mr^{2} = I_{x}  +  I_{y}

\begin{matrix} = 2I_{d} & \boxed{I_{x} = I_{y} \text{ and } I_{z} = \frac{1}{2}Mr^{2}} \end{matrix}

⇒      I_{d} = \frac{1}{4}Mr^{2}.