Perpendicular axis theorem

For any planar (thin, flat) body lying in the xy-plane, the moment of inertia about the z-axis perpendicular to the plane equals the sum of moments about the two in-plane axes: I_z = I_x + I_y. This follows directly from the definition I = ∫ r² dm, noting that for a point in the plane r² = x² + y².

The theorem is a shortcut: compute the easy-in-plane moments, add them, and you have the harder perpendicular-axis moment for free. It only applies to planar bodies — a three-dimensional shape does not in general satisfy it. Combined with the parallel-axis theorem (for shifting to off-centre axes), it handles most textbook moment-of-inertia problems without integration.