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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.