List of moments of inertia

The following is a list of moments of inertia.

Moments of inertia

Moments of inertia have units of dimension mass × length². The following moments of inertia are derived from the fact that the moment of inertia of a point object is ${\displaystyle mr^{2}\,}$ .

Description	Figure	Moment(s) of inertia	Comment
Thin cylindrical shell with open ends, of radius ${\displaystyle r}$  and mass ${\displaystyle m}$		${\displaystyle I=mr^{2}\,}$	—
Thick cylinder with open ends, of inner radius ${\displaystyle r_{1}}$ , outer radius ${\displaystyle r_{2}}$  and mass ${\displaystyle m}$		${\displaystyle I_{z}={\frac {1}{2}}m({r_{1}}^{2}+{r_{2}}^{2})}$  ${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m[3({r_{1}}^{2}+{r_{2}}^{2})+h^{2}]}$  or if we let ${\displaystyle t_{n}}$  be the normalized thickness ${\displaystyle {\frac {t}{r}}}$  and ${\displaystyle r=r_{2}}$  then ${\displaystyle I_{z}=mr^{2}(1-t_{n}+{\frac {1}{2}}t_{n}^{2})}$	—
Solid cylinder of radius ${\displaystyle r}$ , height ${\displaystyle h}$  and mass ${\displaystyle m}$		${\displaystyle I_{z}={\frac {1}{2}}mr^{2}}$  ${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m(3r^{2}+h^{2})}$	—
Thin, solid disk of radius ${\displaystyle r}$  and mass ${\displaystyle m}$		${\displaystyle I_{z}={\frac {1}{2}}mr^{2}}$  ${\displaystyle I_{x}=I_{y}={\frac {1}{4}}mr^{2}}$	—
Solid sphere of radius ${\displaystyle r}$  and mass ${\displaystyle m}$		${\displaystyle I={\frac {2}{5}}mr^{2}}$	—
Hollow sphere of radius ${\displaystyle r}$  and mass ${\displaystyle m}$		${\displaystyle I={\frac {2}{3}}mr^{2}}$	—
Right circular cone with radius ${\displaystyle r}$ , height ${\displaystyle h}$  and mass ${\displaystyle m}$		${\displaystyle I_{z}=(3/10)mr^{2}\,\!}$  ${\displaystyle I_{x}=I_{y}=(3/5)m(r^{2}/4+h^{2})\,\!}$	—
Solid cuboid of height ${\displaystyle h}$ , width ${\displaystyle w}$ , and depth ${\displaystyle d}$ , and mass ${\displaystyle m}$		${\displaystyle I_{h}={\frac {1}{12}}m(w^{2}+d^{2})}$  ${\displaystyle I_{w}={\frac {1}{12}}m(h^{2}+d^{2})}$  ${\displaystyle I_{d}={\frac {1}{12}}m(h^{2}+w^{2})}$	For a similarly oriented cube with sides of length ${\displaystyle s}$  and mass ${\displaystyle M}$ , ${\displaystyle I_{CM}={\frac {1}{6}}ms^{2}}$ .
Rod of length ${\displaystyle L}$  and mass ${\displaystyle m}$		${\displaystyle I_{center}={\frac {1}{12}}mL^{2}}$	This expression is an approximation, and assumes that the mass of the rod is distributed in the form of an infinitely thin (but rigid) wire.
Rod of length ${\displaystyle L}$  and mass ${\displaystyle m}$		${\displaystyle I_{end}={\frac {1}{3}}mL^{2}}$	This expression is an approximation, and assumes that the mass of the rod is distributed in the form of an infinitely thin (but rigid) wire.
Torus of tube radius ${\displaystyle a}$ , cross-sectional radius ${\displaystyle b}$  and mass ${\displaystyle m}$ .		About a diameter:${\displaystyle {\frac {1}{8}}(4a^{2}+5b^{2})m}$  About the vertical axis:${\displaystyle (a^{2}+{\frac {3}{4}}b^{2})m}$
Thin, solid, polygon shaped plate with vertices ${\displaystyle {\vec {P}}_{1}}$ , ${\displaystyle {\vec {P}}_{2}}$ , ${\displaystyle {\vec {P}}_{3}}$ , ..., ${\displaystyle {\vec {P}}_{N}}$  and mass ${\displaystyle m}$ .		${\displaystyle I={\frac {m}{6}}{\frac {\sum _{n=1}^{N}||{\vec {P}}_{n+1}\times {\vec {P}}_{n}||({\vec {P}}_{n+1}^{2}+{\vec {P}}_{n+1}\cdot {\vec {P}}_{n}+{\vec {P}}_{n}^{2})}{\sum _{n=1}^{N}||{\vec {P}}_{n+1}\times {\vec {P}}_{n}||}}}$

Area moments of inertia

The area moment of inertia or second moment of area has a unit of dimension Length⁴. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.

Description	Figure	Area Moment(s) of inertia	Comment
a filled circular area of radius ${\displaystyle r\,}$		${\displaystyle I_{0}=\pi r^{4}/4\,}$
a circular area of inner radius ${\displaystyle r_{1}}$  and outer radius ${\displaystyle r_{2}}$		${\displaystyle I_{0}={\frac {1}{4}}\pi ({r_{2}}^{4}-{r_{1}}^{4})}$
a filled semicircle with radius ${\displaystyle r\,}$  resting on a horizontal line		${\displaystyle I_{0}=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\,}$
a filled semicircle as above but with respect to an axis collinear with the base		${\displaystyle I=\pi r^{4}/8\,}$
a filled semicircle as above but with respect to a vertical axis through the centroid		${\displaystyle I_{0}=\pi r^{4}/8\,}$
a filled quarter circle with radius ${\displaystyle r\,}$  entirely in the 1st quadrant of the Cartesian coordinate system		${\displaystyle I_{0}=\pi r^{4}/16\,}$
a filled ellipse whose radius along the ${\displaystyle x}$ -axis is ${\displaystyle a\,}$  and whose radius along the ${\displaystyle y}$ -axis is ${\displaystyle b\,}$		${\displaystyle I_{0}=\pi ab^{3}/4\,}$
a filled rectangular area with a base width of ${\displaystyle b\,}$  and height ${\displaystyle h\,}$		${\displaystyle I_{0}=bh^{3}/12\,}$
a filled rectangular area as above but with respect to an axis collinear with the base		${\displaystyle I=bh^{3}/3\,}$	This is a trivial result from the parallel axes rule
a filled triangular area with a base width of ${\displaystyle b\,}$  and height ${\displaystyle h}$		${\displaystyle I_{0}=bh^{3}/36\,}$
a filled triangular area as above but with respect to an axis collinear with the base		${\displaystyle I=bh^{3}/12\,}$	This is a consequence of the parallel axes rule and the fact that the distance between these two axes is always ${\displaystyle h/3\,}$
a filled regular hexagon with a side length of ${\displaystyle a\,}$		${\displaystyle I_{0}=5{\sqrt {3}}a^{4}/16\,}$