Centroids and Area Moments of Inertia for 2D Shapes

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 $Area = \pi ab$	$$\begin{align*} I_x &= \frac{\pi}{4} ab^3 \\[4pt] I_y &= \frac{\pi}{4} a^3 b \end{align*}$$