# Centroids and Area Moments of Inertia for 2D Shapes

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Shape with Area and Centroid Location Shown Rectangular Area Moments of Inertia Polar Area Moments of Inertia

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*}

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*}