= Tangent-axis inertia of an axisymmetric lamina
{title2=$I_{\mathrm{tan}}=\frac12I_z+MR^2$}
For an axisymmetric <planar lamina> of total <mass> $M$, let $I_z$ be its <moment of inertia> about the central axis normal to its plane. The <perpendicular axis theorem> and <rotational symmetry> give $I_x=I_y=I_z/2$ about central diameters. An in-plane tangent line a perpendicular distance $R$ from a parallel central diameter therefore has <moment of inertia> $I_{\mathrm{tan}}=I_z/2+MR^2$ by the <parallel axis theorem>. An axis normal to the lamina through the same rim point instead has $I_z+MR^2$; specifying the point without the axis direction is insufficient.
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