List of moments of inertia

The following is a list of moments of inertia.

Area moments of inertia

Area moments of inertia have units of dimension length⁴. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.

Circles and related areas

For a filled circular area of radius $r\,\!$, $I_{0}=\pi r^{4}/4\,\!$.

For a filled semicircle with radius $r\,\!$ resting atop the $x$-axis, $I_{0}=\pi r^{4}/8\,\!$.

For a filled quarter circle with radius $r\,\!$ entirely in the upper-right quadrant of the Cartesian plane, $I_{0}=\pi r^{4}/16\,\!$.

For an ellipse whose radius along the $x$-axis is $a\,\!$ and whose radius along the $y$-axis is $b\,\!$, $I_{0}=\pi ab^{3}/4\,\!$.

Rectangle

area with a base width of $b\,\!$ and height $h\,\!$, $I_{0}=bh^{3}/12\,\!$.

For an axis collinear with the base, $I=bh^{3}/3\,\!$. (This is a trivial result from the parallel axis theorem.)

Triangle

For a filled triangular area with a base width of $b\,\!$ and height $h$, $I_{0}=bh^{3}/36\,\!$.

For an axis collinear with the base, $I=bh^{3}/12\,\!$. (This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is always $h/3\,\!$.)

Mass moments of inertia

Mass moments of inertia have units of dimension mass × length².

Description	Figure	Moment(s) of inertia	Comment
Thin cylindrical shell with open ends, of radius $r$ and mass $m$		$I=mr^{2}\,\!$	—
Thick cylinder with open ends, of inner radius $r_{1}$, outer radius $r_{2}$ and mass $m$		$I={\frac {1}{2}}m({r_{1}}^{2}+{r_{2}}^{2})$	—
Solid cylinder of radius $r$, height $h$ and mass $m$		$I_{z}={\frac {1}{2}}mr^{2}$ $I_{x}=I_{y}={\frac {1}{12}}m(3r^{2}+h^{2})$	—
Thin disk of radius $r$ and mass $m$		$I_{z}={\frac {1}{2}}mr^{2}$ $I_{x}=I_{y}={\frac {1}{4}}m(r^{2})$	—
Solid sphere of radius $r$ and mass $m$		$I={\frac {2}{5}}mr^{2}$	—
Hollow sphere of radius $r$ and mass $m$		$I={\frac {2}{3}}mr^{2}$	—
Right circular cone with radius $r$, $h$ and mass $m$		$I_{z}=(3/10)mr^{2}\,\!$ $I_{x}=I_{y}=(3/5)m(r^{2}/4+h^{2})\,\!$	—
Solid rectangular prism of height $h$, width $w$, and depth $d$, and mass $m$		$I_{h}={\frac {1}{12}}m(w^{2}+d^{2})$ $I_{w}={\frac {1}{12}}m(h^{2}+d^{2})$ $I_{d}={\frac {1}{12}}m(h^{2}+w^{2})$	For a similarly oriented cube with sides of length $s$ and mass $M$, $I_{CM}={\frac {1}{6}}ms^{2}$.
Rod of length $L$ and mass $m$		$I_{center}={\frac {1}{12}}mL^{2}$	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 $L$ and mass $m$		$I_{end}={\frac {1}{3}}mL^{2}$	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.