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List of area moments of inertia: Wikis

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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 Figure Area moment of inertia Comment Reference
a filled circular area of radius r Area moment of inertia of a circle.svg I_0 = \frac{\pi r^4}{4} [1]
an annulus of inner radius r1 and outer radius r2 Area moment of inertia of a circular area.svg 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 Area moment of inertia of a circular sector.svg 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 Area moment of inertia of a semicircle 2.svg 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 Area moment of inertia of a semicircle.svg 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
Area moment of inertia of a semicircle 3.svg
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 Area moment of inertia of a quartercircle.svg 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 Area moment of inertia of a quartercircle 2.svg 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 Area moment of inertia of an ellipsis.svg I_0 = \frac{\pi}{4} ab^3
a filled rectangular area with a base width of b and height h Area moment of inertia of a rectangle.svg I_0 = \frac{bh^3}{12} [4]
a filled rectangular area as above but with respect to an axis collinear with the base Area moment of inertia of a rectangle 2.svg 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 Area moment of inertia of a triangle.svg I_0 = \frac{bh^3}{36} [5]
a filled triangular area as above but with respect to an axis collinear with the base Area moment of inertia of a triangle 2.svg 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 Area moment of inertia of a regular hexagon.svg I_0 = \frac{5\sqrt{3}}{16}a^4 The result is valid for both a horizontal and a vertical axis through the centroid.

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References


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