For a homogeneous (uniformly dense) thin circular disc, the centre of mass lies at:

A homogeneous disc has reflection symmetry about its geometric centre: for every mass element at (x, y) there is an equal element at (−x, −y). The symmetry integrals $\int x \, dm$ and $\int y \, dm$ both vanish about that point, so the COM coincides with the geometric centre. This argument holds for all uniformly dense regular shapes (ring, sphere, rod, etc.).