How to derive the formula for the moment of inertia of a disc about an axis passing through its centre and perpendicular to its plane?

The disc can be considered to be a planar body. Hence the theorem of perpedicular axes is applicable to it.

we take three concurrent axes through the centre of the disc, O as the x,y,z axes ;x and y-axes lie in the plane of the disc and z is perpendicular to it. By the theorem of perpendicular axes,
Iz=Ix+Iy

x and y axes are along two diameters of the disc, and by symmetry the moment of inertia of the disc is the same about any diameter. Hence
Ix=Iy
and Iz=2Ix
But Iz=MR2/2
So finally, Ix=Iz/2=MR2/4

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