Mass Moments Of Inertia Of Common Geometric Shapes

Slender Rod

${\displaystyle I_{x}=0}$

${\displaystyle I_{y}=I_{z}={\frac {1}{12}}ml^{2}}$

Thin Quarter-Circular Rod

${\displaystyle I_{x}=I_{z}=mr^{2}({\frac {1}{2}}-{\frac {4}{\pi ^{2}}})}$

${\displaystyle I_{y}=mr^{2}(1-{\frac {8}{\pi ^{2}}})}$

Thin Ring

${\displaystyle I_{x}=I_{y}={\frac {1}{2}}mr^{2}}$

${\displaystyle I_{z}=mr^{2}}$

Sphere

${\displaystyle I_{x}=I_{y}=I_{z}={\frac {2}{5}}mr^{2}}$

Hemisphere

${\displaystyle I_{x}=I_{y}={\frac {83}{320}}mr^{2}}$

${\displaystyle I_{z}={\frac {2}{5}}mr^{2}}$

Thin Circular Disk

${\displaystyle I_{x}=I_{y}={\frac {1}{4}}mr^{2}}$

${\displaystyle I_{z}={\frac {1}{2}}mr^{2}}$

Rectangular Prism

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

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

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

Right Circular Cylinder

${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m(3r^{2}+h^{2})}$

${\displaystyle I_{z}={\frac {1}{2}}mr^{2}}$

Right Half Cylinder

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

${\displaystyle I_{y}={\frac {1}{12}}mh^{2}+{\frac {1}{4}}mr^{2}}$

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

Thin Rectangular Plate

${\displaystyle I_{x}={\frac {1}{12}}mb^{2}}$

${\displaystyle I_{y}={\frac {1}{12}}ma^{2}}$

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

Right Circular Cone

${\displaystyle I_{x}=I_{y}={\frac {3}{80}}m({4}{r^{2}}+h^{2})}$

${\displaystyle I_{z}={\frac {3}{10}}mr^{2}}$

Right Tetrahedron

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

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

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