# 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 $$\dfrac{2}{3}\dfrac{r\,\sin\alpha}{\alpha}$$ $$2{\alpha}\,r^2$$ $$I_{x^{'}x^{'}} =$$$$\dfrac{r^4}{4} \left(\alpha + \dfrac{1}{2}\sin2\alpha\right)$$

### Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.