The following is a list of area moments of inertia. The area moment of inertia or second moment of area has a unit of dimension length^{4}, 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. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.
Area moments of inertia[edit]
Description  Figure  Area moment of inertia  Comment  Reference 

a filled circular area of radius r  ^{[1]}  
an annulus of inner radius r_{1} and outer radius r_{2}  For thin tubes, and .
We can say that and because this bracket can be simplified to . Ultimately, for a thin tube, . 

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  This formula is valid for only for 0 ≤ ≤  
a filled semicircle with radius r with respect to a horizontal line passing through the centroid of the area  ^{[2]}  
a filled semicircle as above but with respect to an axis collinear with the base  This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is  ^{[2]}  
a filled semicircle as above but with respect to a vertical axis through the centroid 

^{[2]}  
a filled quarter circle with radius r entirely in the 1st quadrant of the Cartesian coordinate system  ^{[3]}  
a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid  This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is  ^{[3]}  
a filled ellipse whose radius along the xaxis is a and whose radius along the yaxis is b  
a filled rectangular area with a base width of b and height h  ^{[4]}  
a filled rectangular area as above but with respect to an axis collinear with the base  This is a result from the parallel axis theorem  ^{[4]}  
a filled rectangular area as above but with respect to an axis collinear, where r is the perpendicular distance from the centroid of the rectangle to the axis of interest  This is a 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  ^{[5]}  
a filled triangular area as above but with respect to an axis collinear with the base  This is a consequence of the parallel axis theorem  ^{[5]}  
a filled regular hexagon with a side length of a  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.  
Any plane region with a known area moment of inertia for a parallel axis. (Main Article parallel axis theorem)  This can be used to determine the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance (r) between the axes. 
See also[edit]
References[edit]
 ^ "Circle". eFunda. Retrieved 20061230.
 ^ ^{a} ^{b} ^{c} "Circular Half". eFunda. Retrieved 20061230.
 ^ ^{a} ^{b} "Quarter Circle". eFunda. Retrieved 20061230.
 ^ ^{a} ^{b} ^{c} "Rectangular area". eFunda. Retrieved 20061230.
 ^ ^{a} ^{b} "Triangular area". eFunda. Retrieved 20061230.
External links[edit]
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