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

Centroid of a Rectangle \[Area=bh\] 		\[I_{x}=\frac{1}{12}bh^{3}\] \[I_{y}=\frac{1}{12}b^{3}h\] \[J_{z}=\frac{1}{12}bh(b^{2}+h^{2})\]

Right Triangle

Centroid of a Right Triangle \[Area=\frac{1}{2}bh\] 		\[I_{x}=\frac{1}{36}bh^{3}\] \[I_{y}=\frac{1}{36}b^{3}h\]
\[I_{x'}=\frac{1}{12}bh^{3}\] \[I_{y'}=\frac{1}{12}b^{3}h\]

Triangle

Centroid of a Triangle \[Area=\frac{1}{2}bh\] 		\[I_{x}=\frac{1}{36}bh^{3}\]
\[I_{x'}=\frac{1}{12}bh^{3}\]

Circle

Centroid of a Circle \[Area=\pi r^{2}\] 		\[I_{x}=\frac{\pi}{4}r^{4}\] \[I_{y}=\frac{\pi}{4}r^{4}\] \[J_{z}=\frac{\pi}{2}r^{4}\]

Circular Annulus

Centroid of a Circular Annulus \[Area=\pi (r_{o}^{2}-r_{i}^{2})\] 		\[I_{x}=\frac{\pi}{4}(r_{o}^{4}-r_{i}^{4})\] \[I_{y}=\frac{\pi}{4}(r_{o}^{4}-r_{i}^{4})\] \[J_{z}=\frac{\pi}{2}(r_{o}^{4}-r_{i}^{4})\]

Semicircle

Centroid of a Semicircle \[Area=\frac{\pi}{2} r^{2}\] 		\[I_{x}=\left(\frac{\pi}{8}-\frac{8}{9\pi}\right) r^{4}\] \[I_{y}=\frac{\pi}{8}r^{4}\]
\[I_{x'}=\frac{\pi}{8}r^{4}\]		 \[J_{z}=\left(\frac{\pi}{4}-\frac{8}{9\pi}\right) r^{4}\]

Quarter Circle

Centroid of a Semicircle \[Area=\frac{\pi}{4} r^{2}\] 		\[I_{x}=\left(\frac{\pi}{16}-\frac{4}{9\pi}\right) r^{4}\] \[I_{y}=\left(\frac{\pi}{16}-\frac{4}{9\pi}\right) r^{4}\]
\[I_{x'}=\frac{\pi}{16}r^{4}\] \[I_{y'}=\frac{\pi}{16}r^{4}\]		 \[J_{z}=\left(\frac{\pi}{8}-\frac{8}{9\pi}\right) r^{4}\]

Ellipse

Centroid of a Ellipse \[Area=\pi ab\] 		\[I_{x}=\frac{\pi}{4}ab^{3}\] \[I_{y}=\frac{\pi}{4}a^{3}b\]

Circular Sector

Centroid of a Circular Sector \[Area=\theta r^2\] 		\[I_{x}=\frac{1}{4} \left(\theta - \frac{1}{2}\sin{2\theta} \right) r^{4}\]

Quarter Circle Arc

Centroid of a Quartercircle Arc \[Length=\frac{\pi}{2} r\]

Semicircle Arc

Centroid of a Semicircle Arc \[Length=\pi r\]

Circular Arc Segment

Centroid of a Circular Arc Segment \[Length=2 \theta r\]

Parabolic Area

Centroid of a Parabolic Area \[Area = \frac{4}{3} ab \]