In-plane inertia isotropy of a square plate ($I_{\varphi }=M{l}^2/12$)

A uniform thin square of side $l$ and mass $M$ has $I_z=M{l}^2/6$ about the perpendicular central axis. Any central in-plane axis and its perpendicular in-plane partner have equal moments by $90^{\circ }$ rotational symmetry. The perpendicular axis theorem says their sum is $I_z$, proving $I_{\varphi }=M{l}^2/12$ for every angle $\varphi$. A non-square rectangle generally lacks this angle independence.