Statics/Geometric Properties of Solids

Mass Moments Of Inertia Of Common Geometric Shapes

Slender Rod

$I_x = 0$

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

Thin Quarter-Circular Rod

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

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

Thin Ring

$I_x = I_y = \frac {1}{2}mr^2$

$I_z = mr^2$

Sphere

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

Hemisphere

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

$I_z = \frac {2}{5} mr^2$

Thin Circular Disk

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

$I_z = \frac {1}{2} mr^2$

Rectangular Prism

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

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

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

Right Circular Cylinder

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

$I_z = \frac {1}{2} mr^2$

Right Half Cylinder

$I_x = \frac {1}{12} mh^2 + mr^2( \frac {1}{4} - \frac {16}{9\pi^2})$

$I_y = \frac {1}{12} mh^2 + \frac {1}{4}mr^2$

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

Thin Rectangular Plate

$I_x = \frac {1}{12} mb^2$

$I_y = \frac {1}{12} ma^2$

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

Right Circular Cone

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

$I_z = \frac {3}{10} mr^2$

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