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List of second moments of area

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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]
anannulus of inner radius r₁ and outer radius r₂	$I_{0}={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)$	For thin tubes, this is approximately equal to: $\pi \left({\frac {{r_{2}}+{r_{1}}}{2}}\right)^{3}\left({r_{2}}-{r_{1}}\right)$ or $\pi$ times the cube of the average radius times the thickness.
a filled circular sector of angle θinradians and radius r with respect to an axis through the centroid of the sector and the centre of the circle	$I_{0}=\left(\theta -{\frac {1}{2}}\sin 2\theta \right){\frac {r^{4}}{8}}$
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 axis theorem 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 axis theorem 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 axis theorem	^[4]
a filled triangular area with a base width of b and height h with respect to an axis through the centroid	$I_{0}={\frac {bh^{3}}{36}}$		^[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 axis theorem	^[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.