Inertias for common shapes

Rectangular parallelepiped/prism (a box)

with length $a$, width $b$, height $c$

$$V=abc$$

$$I_{xx}=\frac{1}{12}m(b^2+c^2)$$ $$I_{yy}=\frac{1}{12}m(a^2+c^2)$$ $$I_{zz}=\frac{1}{12}m(a^2+b^2)$$

$$I_{xy}=I_{yz}=I_{zx}=0$$

Solid Cylinder

with radius $r$ and length $h$

$$V=\pi r^2 h$$

$$I_{xx}=I_{yy}=\frac{1}{12}m(3r^2+h^2)$$ $$I_{zz}=\frac{1}{2}m(r^2)$$ $$I_{xy}=I_{yz}=I_{zx}=0$$

Thick-walled cylinder

with inner radius $r_i$ and outer radius $r_o$ and length $h$

$$V=\pi (r_o^2-r_i^2) h$$

$$I_{xx}=I_{yy}=\frac{1}{12}m(3(r_o^2+r_i^2)+h^2)$$ $$I_{zz}=\frac{1}{2}m(r_o^2+r_i^2)$$ $$I_{xy}=I_{yz}=I_{zx}=0$$

Thin-walled cylinder

with radius $r$ and length $h$

$$I_{xx}=I_{yy}=\frac{1}{12}m(6r^2+h^2)$$ $$I_{zz}=m(r^2)$$ $$I_{xy}=I_{yz}=I_{zx}=0$$

Solid Sphere

with radius $r$

$$V=\frac{4}{3}\pi r^3$$

$$I_{xx}=I_{yy}=I_{zz}=\frac{2}{5}mr^2$$ $$I_{xy}=I_{yz}=I_{zx}=0$$

Thin Hollow Sphere

with radius $r$

$$I_{xx}=I_{yy}=I_{zz}=\frac{2}{3}mr^2$$ $$I_{xy}=I_{yz}=I_{zx}=0$$

Solid Ellipsoid

with axis lengths $a$, $b$, and $c$

$$V=\frac{4}{3}\pi abc $$

$$I_{xx}=\frac{1}{5}m(b^2+c^2)$$ $$I_{yy}=\frac{1}{5}m(a^2+c^2)$$ $$I_{zz}=\frac{1}{5}m(a^2+b^2)$$

$$I_{xy}=I_{yz}=I_{zx}=0$$

Thin Hollow Ellipsoid

with axis lengths $a$, $b$, and $c$

$$I_{xx}=\frac{1}{3}m(b^2+c^2)$$ $$I_{yy}=\frac{1}{3}m(a^2+c^2)$$ $$I_{zz}=\frac{1}{3}m(a^2+b^2)$$

$$I_{xy}=I_{yz}=I_{zx}=0$$