# 3.6.1: Tables of geometries


The following tables present several moment of inertias of commonly used geometries.

Table 3.1 Moments of Inertia for various plane surfaces about their center of gravity (full shapes).

 Shape Name Picture Description $$\mathbf{x_c}$$, $$\mathbf{x_c}$$ $$\mathbf{A}$$ $$\mathbf{I_{xx}}$$ Rectangle $$\dfrac{a}{2}\,;\dfrac{b}{2}$$ $$a\,b$$ $$\dfrac{a\,b^{3}}{12}$$ Triangle $$\dfrac{a}{3}$$ $$\dfrac{a\,b}{3}$$ $$\dfrac{a\,b^{3}}{36}$$ Circle $$\dfrac{b}{2}$$ $$\dfrac{\pi\, b^2}{4}$$ $$\dfrac{\pi b^4}{64}$$ Ellipse $$\dfrac{a}{2}\; \dfrac{b}{2}$$ $$\dfrac{\pi\, ab}{4}$$ $$\dfrac{ab^2}{64}$$ $$y=\alpha\,x^2$$ Parabola $$\dfrac{3\,\alpha\,b}{15\,\alpha-5}$$ $$\dfrac{6\alpha -2}{3} \, \left( \dfrac{b}{\alpha}\right)^{\dfrac{3}{2}}$$ $$\dfrac{\sqrt{b}\,\left( 20\,{b}^{3}-14\,{b}^{2}\right) }{35\,\sqrt{\alpha}}$$ Quadrant of Circle $$\dfrac{4\,r}{3\,\pi}$$ $$\dfrac{\pi\,r^2}{4}$$ $$r^4\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)$$ Ellipsoidal Quadrant $$\dfrac{4\,b}{3\,\pi}$$ $$\dfrac{\pi\,a\,b}{4}$$ $$a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)$$ half of of Ellipse $$\dfrac{4\,b}{3\,\pi}$$ $$\dfrac{\pi\,a\,b}{4}$$ $$a\,b^3\left(\dfrac{\pi}{16} - \dfrac{4}{9\pi}\right)$$ Circular Sector $$0$$ $$2{\alpha}\,r^2$$ $$\dfrac{r^4}{4} \left(\alpha - \dfrac{1}{2}\sin2\alpha\right)$$ Circular Sector $$\dfrac{2}{3}\dfrac{r\,\sin\alpha}{\alpha}$$ $$2{\alpha}\,r^2$$ $$I_{x^{'}x^{'}} =$$$$\dfrac{r^4}{4} \left(\alpha + \dfrac{1}{2}\sin2\alpha\right)$$