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