Moment of inertia of sphere

Question

I know that the moment of inertia of a sphere is $\frac{2}{5}mr^2$ but using the perpendicular axis theorem we can get that the $I_z=I_x+I_y$ and they are all equal in case of the sphere because of its symmetry, therefore $I = I + I$ and then $I=0$! Isn't that wrong?

Answer 1

The perpendicular axis theorem can be only used to determine the moment of inertia of a rigid object that lies entirely within a plane. And a sphere does not.