Skip to main content
Logo image

Engineering Statics: Open and Interactive

Section C.2 Moment of Inertia of Common Shapes

Table C.2.1. Moments of Inertia of Common Shapes
Shape Centroid Centroidal MOI \(I_x, \ I_y\)
Rectangle with base b and height h
\((b/2, h/2)\)
\(\bar{I}_{x'} = \dfrac{1}{12} b h^3\)
\(\bar{I}_{y'} = \dfrac{1}{12} b^3 h\)
\(I_{x} = \dfrac{1}{3} b h^3\)\(I_{y} = \dfrac{1}{3} b^3 h\)
Right triangle with vertices (0,0), (0, h), and (b, 0)
\((b/3, h/3)\)
\(\bar{I}_{x'} = \dfrac{1}{36} b h^3\)\(\bar{I}_{y'} = \dfrac{1}{36} b^3 h\)
\(I_x = \dfrac{1}{12} b h^3\)\(I_y = \dfrac{1}{12} b^3 h\)
Circle with centered at *x-bar, y-bar). The x’ and y’ axes intersect the centroid.
\((r,r)\)
\(\bar{I}_{x'}=\bar{I}_{y'}= \dfrac{\pi}{4} r^4\)
\(I_{x}=I_{y}= \dfrac{5 \pi}{4} r^4\)
Half circle with base on x-axis and centered on y-axis. The x’ axis intersects the centroid.
\(\left (r, \dfrac{4r}{3\pi} \right)\)
\(\bar{I}_{x'} = \left(\frac{\pi}{8} - \frac{8}{9\pi}\right) r^4\)
\(\bar{I}_{x'} \approx 0.1098\ r^4\)
\(\bar{I}_{y'} = \dfrac{\pi}{8} r^4 \)
\(I_x =\dfrac{\pi}{8} r^4\)
\(I_y =\dfrac{5 \pi}{8} r^4\)
Quarter circle with base on x-axis and side on y-axis. The x’ axis intersects the centroid.
\(\left (\dfrac{4r}{3\pi}, \dfrac{4r}{3\pi} \right)\)
\(\bar{I}_{x'} = \frac{1}{2}\left(\frac{\pi}{8} - \frac{8}{9\pi}\right) r^4 \)
\(\bar{I}_{x'} = \bar{I}_{y'} \approx 0.0549\ r^4\)
\(I_x = I_y = \dfrac{\pi }{16}r^4\)