whose centroid is located at the origin. represents the second moment of area with respect to the x-axis; represents the second moment of area with respect to the y-axis; represents the polar moment of inertia with respect to the z-axis. Using the perpendicular axis theorem we get the value of . Annulus centered at origin Consider an annulus whose center is at the origin, outside radius is , and inside radius is . Because of the symmetry of the annulus, the centroid also lies at the origin. We can determine the polar moment of inertia, , about the axis by the method of composite shapes. This polar moment of inertia is equivalent to the polar moment of inertia of a circle with radius minus the polar moment of inertia of a circle with radius , both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius with respect to the origin. In this case, it is easier to directly calculate as we already have , which has both an and component. Instead of obtaining the second moment of area from Cartesian coordinates as done in the previous section, we shall calculate and directly using polar coordinates. Now, the polar moment of inertia about the axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius and a circle with radius . Alternatively, we could change the limits on the integral the first time around to reflect the fact that there is a hole. This would be done like this. Any polygon The second moment of area about the origin for any simple polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon after dividing the area into a set of triangles. This formula is related to the shoelace formula and can be considered a special case of Green's theorem. A polygon is assumed to have vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct. where are the coordinates of the -th polygon vertex, for . Also, are assumed to be equal to the coordinates of the first vertex, i.e., and . See also List of second moments of area List of moments of inertia Moment of inertia Parallel axis theorem