See
list of second moments of area for other shapes.
Rectangle with centroid at the origin Consider a rectangle with base b and height h whose
centroid is located at the origin. I_x represents the second moment of area with respect to the x-axis; I_y represents the second moment of area with respect to the y-axis; J_z represents the polar moment of inertia with respect to the z-axis. \begin{align} I_x &= \iint_{R} y^2\, dA = \int^\frac{b}{2}_{-\frac{b}{2}} \int^\frac{h}{2}_{-\frac{h}{2}} y^2 \,dy \,dx = \int^\frac{b}{2}_{-\frac{b}{2}} \frac{1}{3}\frac{h^3}{4}\,dx = \frac{b h^3}{12} \\ I_y &= \iint_{R} x^2\, dA = \int^\frac{b}{2}_{-\frac{b}{2}} \int^\frac{h}{2}_{-\frac{h}{2}} x^2 \,dy \,dx = \int^\frac{b}{2}_{-\frac{b}{2}} h x^2\, dx = \frac{b^3 h}{12} \end{align} Using the
perpendicular axis theorem we get the value of J_z. J_z = I_x + I_y = \frac{b h^3}{12} + \frac{h b^3}{12} = \frac{b h}{12}\left(b^2 + h^2\right)
Annulus centered at origin Consider an
annulus whose center is at the origin, outside radius is r_2, and inside radius is r_1. Because of the symmetry of the annulus, the
centroid also lies at the origin. We can determine the polar moment of inertia, J_z, about the z 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 r_2 minus the polar moment of inertia of a circle with radius r_1, both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius r with respect to the origin. In this case, it is easier to directly calculate J_z as we already have r^2, which has both an x and y component. Instead of obtaining the second moment of area from
Cartesian coordinates as done in the previous section, we shall calculate I_x and J_z directly using
polar coordinates. \begin{align} I_{x, \text{circle}} &= \iint_{R} y^2\,dA = \iint_{R} \left(r\sin{\theta}\right)^2\, dA = \int_0^{2\pi}\int_0^r \left(r\sin{\theta}\right)^2\left(r \, dr \, d\theta\right) \\ &= \int_0^{2\pi}\int_0^r r^3\sin^2{\theta}\,dr \, d\theta = \int_0^{2\pi} \frac{r^4\sin^2{\theta}}{4}\,d\theta = \frac{\pi}{4}r^4 \\ J_{z, \text{circle}} &= \iint_{R} r^2\, dA = \int_0^{2\pi}\int_0^r r^2\left(r\,dr\,d\theta\right) = \int_0^{2\pi}\int_0^r r^3\,dr\,d\theta \\ &= \int_0^{2\pi} \frac{r^4}{4}\,d\theta = \frac{\pi}{2}r^4 \end{align} Now, the polar moment of inertia about the z axis for an annulus is simply, as stated above, the difference of the second moments of
area of a circle with radius r_2 and a circle with radius r_1. J_z = J_{z, r_2} - J_{z, r_1} = \frac{\pi}{2}r_2^4 - \frac{\pi}{2}r_1^4 = \frac{\pi}{2}\left(r_2^4 - r_1^4\right) Alternatively, we could change the limits on the dr integral the first time around to reflect the fact that there is a hole. This would be done like this. \begin{align} J_{z} &= \iint_{R} r^2 \, dA = \int_0^{2\pi}\int_{r_1}^{r_2} r^2\left(r\, dr\, d\theta\right) = \int_0^{2\pi}\int_{r_1}^{r_2} r^3\, dr\, d\theta \\ &= \int_0^{2\pi}\left[\frac{r_2^4}{4} - \frac{r_1^4}{4}\right]\, d\theta = \frac{\pi}{2}\left(r_2^4 - r_1^4\right) \end{align}
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 n vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct. \begin{align} I_y &= \frac{1}{12}\sum_{i=1}^{n} \left( x_i y_{i+1} - x_{i+1} y_i\right)\left( x_i^2 + x_i x_{i+1} + x_{i+1}^2 \right) \\ I_x &= \frac{1}{12}\sum_{i=1}^{n} \left( x_i y_{i+1} - x_{i+1} y_i\right)\left( y_i^2 + y_i y_{i+1} + y_{i+1}^2 \right) \\ I_{xy} &= \frac{1}{24}\sum_{i=1}^{n} \left( x_i y_{i+1} - x_{i+1} y_i\right) \left( x_i y_{i+1} + 2 x_i y_i + 2 x_{i+1} y_{i+1} + x_{i+1} y_i \right) \end{align} where x_i,y_i are the coordinates of the i-th polygon vertex, for 1 \le i \le n. Also, x_{n+1}, y_{n+1} are assumed to be equal to the coordinates of the first vertex, i.e., x_{n+1} = x_1 and y_{n+1} = y_1. ==See also==