Statics/Geometric Properties of Solids

Mass Moments Of Inertia Of Common Geometric Shapes

Slender Rod

$I_x=0$

$I_y=I_z=\frac{1}{12}m{l}^2$

Thin Quarter-Circular Rod

$I_x=I_z=m{r}^2(\frac{1}{2}-\frac{4}{{\pi }^2})$

$I_y=m{r}^2(1-\frac{8}{{\pi }^2})$

Thin Ring

$I_x=I_y=\frac{1}{2}m{r}^2$

$I_z=m{r}^2$

Sphere

$I_x=I_y=I_z=\frac{2}{5}m{r}^2$

Hemisphere

$I_x=I_y=\frac{83}{320}m{r}^2$

$I_z=\frac{2}{5}m{r}^2$

Thin Circular Disk

$I_x=I_y=\frac{1}{4}m{r}^2$

$I_z=\frac{1}{2}m{r}^2$

Rectangular Prism

$I_x=\frac{1}{12}m(b^2+c^2)$

$I_y=\frac{1}{12}m(a^2+c^2)$

$I_z=\frac{1}{12}m(a^2+b^2)$

Right Circular Cylinder

$I_x=I_y=\frac{1}{12}m(3r^2+h^2)$

$I_z=\frac{1}{2}m{r}^2$

Right Half Cylinder

$I_x=\frac{1}{12}m{h}^2+m{r}^2(\frac{1}{4}-\frac{16}{9{\pi }^2})$

$I_y=\frac{1}{12}m{h}^2+\frac{1}{4}m{r}^2$

$I_z=m{r}^2(\frac{1}{2}-\frac{16}{9{\pi }^2})$

Thin Rectangular Plate

$I_x=\frac{1}{12}m{b}^2$

$I_y=\frac{1}{12}m{a}^2$

$I_z=\frac{1}{12}m(a^2+b^2)$

Right Circular Cone

$I_x=I_y=\frac{3}{80}m(4r^2+h^2)$

$I_z=\frac{3}{10}m{r}^2$

Right Tetrahedron

$I_x=\frac{3}{80}m(b^2+c^2)$

$I_y=\frac{3}{80}m(a^2+c^2)$

$I_z=\frac{3}{80}m(a^2+b^2)$

further reading

• Statics/Geometric_Properties_of_Lines_and_Areas
• Mechanics/Rigid Bodies