Centroids and Area Moments of Inertia for 2D Shapes
- Page ID
- 58050
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Shape with Area and Centroid Location Shown | Rectangular Area Moments of Inertia | Polar Area Moments of Inertia |
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Rectangle \(Area = bh\) |
\begin{align*} I_x &= \frac{1}{12} b h^3 \\[4pt] I_y &= \frac{1}{12} b^3 h \end{align*} |
\( J_z = \dfrac{1}{12} bh(b^2 + h^2) \) |
Right Triangle \(Area = \dfrac{1}{2} bh\) |
\begin{align*} I_x &= \frac{1}{36} bh^3 \\[4pt] I_y &= \frac{1}{36} b^3 h \end{align*} \begin{align*} I_{x'} &= \frac{1}{12} bh^3 \\[4pt] I_{y'} &= \frac{1}{12} b^3 h \end{align*} |
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Triangle \(Area = \dfrac{1}{2} bh\) |
\(I_x = \dfrac{1}{36} bh^3\) \(I_{x'} = \dfrac{1}{12} bh^3\) |
|
Circle \(Area = \pi r^2\) |
\begin{align*} I_x &= \frac{\pi}{4} r^4\\[4pt] I_y &= \frac{\pi}{4} r^4 \end{align*} |
\(J_z = \dfrac{\pi}{2} r^4\) |
Circular Annulus \(Area = \pi (r_o^2 - r_i^2) \) |
\begin{align*} I_x &= \frac{\pi}{4} (r_o^4 - r_i^4) \\[4pt] I_y &= \frac{\pi}{4} (r_o^4 - r_i^4) \end{align*} |
\(J_z = \dfrac{\pi}{2} (r_o^4 - r_i^4)\) |
Semicircle \(Area = \dfrac{\pi}{2} r^2\) |
\(I_x = \left( \dfrac{\pi}{8} - \dfrac{8}{9 \pi} \right) r^4 \) \(I_y = \dfrac{\pi}{8} r^4\) \(I_{x'} = \dfrac{\pi}{8} r^4\) |
\(J_z = \left( \dfrac{\pi}{4} - \dfrac{8}{9 \pi} \right) r^4\) |
Quarter Circle \(Area = \dfrac{\pi}{4} r^2\) |
\begin{align*} I_x &= \left( \frac{\pi}{16} - \frac{4}{9 \pi} \right) r^4 \\[4pt] I_y &= \left( \frac{\pi}{16} - \frac{4}{9 \pi} \right) r^4 \end{align*} \begin{align*} I_{x'} &= \frac{\pi}{16} r^4 \\[4pt] I_{y'} &= \frac{\pi}{16} r^4 \end{align*} |
\( J_z = \left( \dfrac{\pi}{8} - \dfrac{8}{9 \pi} \right) r^4 \) |
Ellipse \(Area = \pi ab\) |
\begin{align*} I_x &= \frac{\pi}{4} ab^3 \\[4pt] I_y &= \frac{\pi}{4} a^3 b \end{align*} |