Perpendicular axis theorem

For a planar body, the moment of inertia about an axis perpendicular to its plane equals the sum of the moments about two perpendicular in-plane axes through the same point.

Moment of inertia: A measure of how mass is distributed relative to an axis, determining resistance to angular acceleration. The theorem relates three moments of inertia defined about axes through one point.
Planar body: An idealized body whose mass or area lies entirely in a single plane. The theorem requires the body's points to share one plane.
Thin plate: A flat body whose thickness is small compared with its other dimensions. A thin plate approximates the planar mass distribution to which the theorem applies.
Three-dimensional rigid body: An extended object whose mass occupies volume in three-dimensional space. Out-of-plane mass adds a squared-distance term, so the planar equality does not generally hold.
Radius of gyration: The distance from an axis at which an object's entire mass could be concentrated without changing its moment of inertia. It expresses each axis moment as total mass times a squared characteristic distance.
Axis of rotation: A line about which rotational motion or rotational quantities are defined. Each moment in the theorem is measured relative to an axis.
Circular disk: A flat, circular body with uniform or specified mass distribution. Its symmetry makes the two perpendicular in-plane moments equal, simplifying the perpendicular moment.
Steiner's theorem: A geometric theorem stating that a region's polar second moment about an exterior point exceeds the centroidal value by area times squared distance. It concerns translating the pole of an area moment, not combining perpendicular axes at one point.
Parallel axis theorem: A theorem that relates an object's moment of inertia about an axis to that about a parallel axis through its center of mass. It shifts axes between points, whereas the perpendicular axis theorem relates directions at one point.
Perpendicular lines: Lines that intersect at a right angle. The two in-plane axes must be perpendicular to each other and to the out-of-plane axis.

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