Appendix E — Centroids and Area Moment of Inertia for Common Shapes

E.1 Centroid coordinates for common shapes

Shape	\(\bar{x}\)	\(\bar{y}\)	Area
	\(\frac{b}{3}\)	\(\frac{h}{3}\)	\(\frac{1}{2}bh\)
	\(\frac{4r}{3\pi}\)	\(\frac{4r}{3\pi}\)	\(\frac{1}{4}\pi r^2\)
	r	\(\frac{4r}{3\pi}\)	\(\frac{1}{2}\pi r^2\)
	\(\frac{3}{4}b\)	\(\frac{3}{10}h\)	\(\frac{1}{3}ah\)

E.2 Area Moment of Inertia for common shapes

Shape	Ix	Iy	J
	\(I_x=\frac{bh^3}{3}\) \(\overline{I}_{x'}=\frac{bh^3}{12}\)	\(I_y=\frac{hb^3}{3}\) \(\overline{I}_{y'}=\frac{hb^3}{12}\)	\(\overline{J}_C=\frac{bh}{12}(b^2+h^2)\)
	\(I_x=\frac{bh^3}{12}\) \(\overline{I}_{x'}=\frac{bh^3}{36}\)	\(I_y=\frac{hb^3}{12}\) \(\overline{I}_{y'}=\frac{hb^3}{36}\)	\(\overline{J}_C=\frac{bh}{36}(b^2+h^2)\)
	\(\overline{I}_{x'}=\frac{\pi r^4}{4}\)	\(\overline{I}_{y'}=\frac{\pi r^4}{4}\)	\(\overline{J}_C=\frac{\pi r^4}{2}\)
	\(I_x=\frac{\pi r^4}{8}\)	\(I_y=\frac{\pi r^4}{8}\)	\(J_{base}=\frac{\pi r^4}{4}\)

References

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All figures in this appendix were created by Kindred Grey in 2025 and released under a CC BY-NC-SA license.