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List of second moments of area

## Summary

The following is a list of second moments of area of some shapes. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The unit of dimension of the second moment of area is length to fourth power, L4, and should not be confused with the mass moment of inertia. If the piece is thin, however, the mass moment of inertia equals the area density times the area moment of inertia.

## Second moments of area

Please take into account that in the following equations,

${\displaystyle I_{x}=\iint _{A}y^{2}\,dx\,dy}$

and
${\displaystyle I_{y}=\iint _{A}x^{2}\,dx\,dy.}$

Description Figure Second moment of area Comment
A filled circular area of radius r   ${\displaystyle I_{x}={\frac {\pi }{4}}r^{4}}$

${\displaystyle I_{y}={\frac {\pi }{4}}r^{4}}$

${\displaystyle I_{z}={\frac {\pi }{2}}r^{4}}$  [1]
${\displaystyle I_{z}}$  is the Second polar moment of area.
An annulus of inner radius r1 and outer radius r2   ${\displaystyle I_{x}={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)}$

${\displaystyle I_{y}={\frac {\pi }{4}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)}$

${\displaystyle I_{z}={\frac {\pi }{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)}$
For thin tubes, ${\displaystyle r\equiv r_{1}\approx r_{2}}$  and ${\displaystyle r_{2}\equiv r_{1}+t}$ . So, for a thin tube, ${\displaystyle I_{x}=I_{y}\approx \pi {r}^{3}{t}}$ .

${\displaystyle I_{z}}$  is the Second polar moment of area.
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 center of the circle   ${\displaystyle I_{x}=\left(\theta -\sin \theta \right){\frac {r^{4}}{8}}}$  This formula is valid only for 0 ≤ ${\displaystyle \theta }$ ${\displaystyle 2\pi }$
A filled semicircle with radius r with respect to a horizontal line passing through the centroid of the area   ${\displaystyle I_{x}=\left({\frac {\pi }{8}}-{\frac {8}{9\pi }}\right)r^{4}\approx 0.1098r^{4}}$

${\displaystyle I_{y}={\frac {\pi r^{4}}{8}}}$  [2]
A filled semicircle as above but with respect to an axis collinear with the base   ${\displaystyle I_{x}={\frac {\pi r^{4}}{8}}}$

${\displaystyle I_{y}={\frac {\pi r^{4}}{8}}}$  [2]
${\displaystyle I_{x}}$ : This is a consequence of the parallel axis theorem and the fact that the distance between the x axes of the previous one and this one is ${\displaystyle {\frac {4r}{3\pi }}}$
A filled quarter circle with radius r with the axes passing through the bases   ${\displaystyle I_{x}={\frac {\pi r^{4}}{16}}}$

${\displaystyle I_{y}={\frac {\pi r^{4}}{16}}}$  [3]
A filled quarter circle with radius r with the axes passing through the centroid   ${\displaystyle I_{x}=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}}$

${\displaystyle I_{y}=\left({\frac {\pi }{16}}-{\frac {4}{9\pi }}\right)r^{4}\approx 0.0549r^{4}}$  [3]
This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is ${\displaystyle {\frac {4r}{3\pi }}}$
A filled ellipse whose radius along the x-axis is a and whose radius along the y-axis is b  ${\displaystyle I_{x}={\frac {\pi }{4}}ab^{3}}$

${\displaystyle I_{y}={\frac {\pi }{4}}a^{3}b}$
A filled rectangular area with a base width of b and height h   ${\displaystyle I_{x}={\frac {bh^{3}}{12}}}$

${\displaystyle I_{y}={\frac {b^{3}h}{12}}}$  [4]
A filled rectangular area as above but with respect to an axis collinear with the base   ${\displaystyle I_{x}={\frac {bh^{3}}{3}}}$

${\displaystyle I_{y}={\frac {b^{3}h}{3}}}$  [4]
This is a result from the parallel axis theorem
A hollow rectangle with an inner rectangle whose width is b1 and whose height is h1  ${\displaystyle I_{x}={\frac {bh^{3}-b_{1}{h_{1}}^{3}}{12}}}$

${\displaystyle I_{y}={\frac {b^{3}h-{b_{1}}^{3}h_{1}}{12}}}$
A filled triangular area with a base width of b, height h and top vertex displacement a, with respect to an axis through the centroid

${\displaystyle I_{x}={\frac {bh^{3}}{36}}}$

${\displaystyle I_{y}={\frac {b^{3}h-b^{2}ha+bha^{2}}{36}}}$  [5]
A filled triangular area as above but with respect to an axis collinear with the base

${\displaystyle I_{x}={\frac {bh^{3}}{12}}}$

${\displaystyle I_{y}={\frac {b^{3}h+b^{2}ha+bha^{2}}{12}}}$  [5]
This is a consequence of the parallel axis theorem
An equal legged angle, commonly found in engineering applications   ${\displaystyle I_{x}=I_{y}={\frac {t(5L^{2}-5Lt+t^{2})(L^{2}-Lt+t^{2})}{12(2L-t)}}}$

${\displaystyle I_{(xy)}={\frac {L^{2}t(L-t)^{2}}{4(t-2L)}}}$

${\displaystyle I_{a}={\frac {t(2L-t)(2L^{2}-2Lt+t^{2})}{12}}}$

${\displaystyle I_{b}={\frac {t(2L^{4}-4L^{3}t+8L^{2}t^{2}-6Lt^{3}+t^{4})}{12(2L-t)}}}$
${\displaystyle I_{(xy)}}$  is the often unused "product second moment of area", used to define principal axes
A filled regular hexagon with a side length of a   ${\displaystyle I_{x}={\frac {5{\sqrt {3}}}{16}}a^{4}}$

${\displaystyle I_{y}={\frac {5{\sqrt {3}}}{16}}a^{4}}$
The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.

## Parallel axis theorem

The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (d) between the axes.

${\displaystyle I_{x'}=I_{x}+Ad^{2}}$