List of area moments of inertia

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The following is list of area moments of inertia. The area moment of inertia or second moment of area has a unit of dimension length⁴, and should not be confused with the mass moment of inertia. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.

Description	Area moment of inertia	Comment	Reference
a filled circular area of radius $r \,$	$I_0 = \frac{\pi r^4}{4} \,$		^[1]
an annulus of inner radius $r 1$ and outer radius $r 2$	$I_0 = \frac{1}{4} \pi\left({r_2}^4-{r_1}^4\right)$
a filled circular sector of angle $\theta \,$ in radians and radius $r \,$ with respect to an axis through the centroid of the sector and the centre of the circle	$I_0 = \left(\theta -\sin\theta\right)\frac{r^4}{8} \,$
a filled semicircle with radius $r \,$ with respect to a horizontal line passing through the centroid of the area	$I_0 = \left(\frac{\pi}{8} - \frac{8}{9\pi}\right)r^4 \,$		^[2]
a filled semicircle as above but with respect to an axis collinear with the base	$I = \frac{\pi r^4}{8} \,$	This is a consequence of the parallel axes rule and the fact that the distance between these two axes is $\frac{4r}{3\pi} \,$	^[2]
a filled semicircle as above but with respect to a vertical axis through the centroid	$I_0 = \frac{\pi r^4}{8} \,$		^[2]
a filled quarter circle with radius $r \,$ entirely in the 1st quadrant of the Cartesian coordinate system	$I = \frac{\pi r^4}{16} \,$		^[3]
a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid	$I_0 = \left(\frac{\pi}{16}-\frac{4}{9\pi}\right)r^4 \,$	This is a consequence of the parallel axes rule and the fact that the distance between these two axes is $\frac{4r}{3\pi} \,$	^[3]
a filled ellipse whose radius along the $x$ -axis is $a \,$ and whose radius along the $y$ -axis is $b \,$	$I_0 = \frac{\pi}{4} ab^3 \,$
a filled rectangular area with a base width of $b \,$ and height $h \,$	$I_0 = \frac{bh^3}{12} \,$		^[4]
a filled rectangular area as above but with respect to an axis collinear with the base	$I = \frac{bh^3}{3} \,$	This is a trivial result from the parallel axes rule	^[4]
a filled triangular area with a base width of $b \,$ and height $h$	$I_0 = \frac{bh^3}{36} \,$		^[5]
a filled triangular area as above but with respect to an axis collinear with the base	$I = \frac{bh^3}{12} \,$	This is a consequence of the parallel axes rule and the fact that the distance between these two axes is always $\frac{h}{3} \,$	^[5]
a filled regular hexagon with a side length of $a \,$	$I_0 = \frac{5\sqrt{3}}{16}a^4 \,$	The result is valid for both a horizontal and a vertical axis through the centroid.