# 3.6.1: Tables of geometries

- Page ID
- 656

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 and Attributions

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.