Perpendicular Axis Theorem

The perpendicular axis theorem applies to flat planar laminae and states $I_z=I_x+I_y$, where the $z$-axis is perpendicular to the plate and the $x$- and $y$-axes lie in its plane through the same point. By the symmetry of a square plate, the in-plane moments of inertia about the two central axes parallel to its edges are equal, $I_x=I_y=I$. Therefore $I_z=I+I=2I$. The value I forgets that two separate in-plane contributions must be summed. The value I/2 inverts the relationship between the perpendicular and in-plane axes. The value 4I wrongly squares the count of contributions instead of adding them. This is the NCERT perpendicular axis theorem, valid only for two-dimensional bodies of negligible thickness. As a consistency check, the perpendicular axis sees mass distributed in both planar directions at once, so its moment of inertia must be larger than either single in-plane value, and $2I>I$ confirms this expectation cleanly. The theorem holds only for thin laminae because it assumes every mass element lies in the plane of the plate; for a thick three-dimensional block the mass spreads along the perpendicular axis itself and the simple sum $I_x+I_y$ no longer applies.