# List of area moments of inertia: Wikis

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# Encyclopedia

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 Figure 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 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 π times the cube of the average radius times the thickness.
a filled circular sector of angle θ 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 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.

## References

1. ^ "Circle". eFunda. Retrieved 2006-12-30.
2. ^ a b c "Circular Half". eFunda. Retrieved 2006-12-30.
3. ^ a b "Quarter Circle". eFunda. Retrieved 2006-12-30.
4. ^ a b "Rectangular area". eFunda. Retrieved 2006-12-30.
5. ^ a b "Triangular area". eFunda. Retrieved 2006-12-30.