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Engineering Statics: Open and Interactive

Appendix C Properties of Shapes

Table C.0.1. Centroids of Common Shapes
Shape
Area
\(\bar{x}\)
\(\bar{y}\)
A rectangle with width of b and height of h, centroid at (x-bar,y-bar).
\(A = b h\)
\(b/2\)
\(h/2\)
A right triangle with right angle at the origin. Width of b and height of h, centroid at (x-bar,y-bar).
\(\dfrac{bh}{2}\)
\(b/3\)
\(h/3\)
A trapezoid with bottom width of b, top width of a, and height of h, centroid at (x-bar,y-bar).
\(\dfrac{(a+b) h}{2}\)
\(\dfrac{a^2 +ab + b^2}{3 (a+b)}\)
\(\dfrac{h(2a+b)}{3(a+b)}\)
A circle with radius of r, centroid at (x-bar,y-bar)=(r,r).
\(\pi r^2\)
\(r\)
\(r \)
A semicircle with radius of r, centroid at (x-bar,y-bar)=(r,4r/(3*pi)).
\(\dfrac{\pi r^2}{2}\)
\(r\)
\(\dfrac{4 r}{3 \pi}\)  1 
A quarter circle with radius of r. The center of curvature is at the origin. Centroid at (x-bar,y-bar)=(4r/(3*pi),4r/(3*pi)).
\(\dfrac{\pi r^2}{4}\) \(\dfrac{4 r}{3 \pi}\) \(\dfrac{4 r}{3 \pi}\)
Table C.0.2. 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.
\(\left (r, \dfrac{4r}{3\pi} \right)\)
\(\bar{I}_{x'} = \left(\dfrac{\pi}{8} - \dfrac{8}{9\pi}\right) r^4\)
\(\bar{I}_{x'} \approx 0.1098\ r^4\)
\(\bar{I}_{y'} = \dfrac{\pi}{8} r^4 \)
\(I_x =\dfrac{\pi}{8} r^4\)
\(I_y =\dfrac{5 \pi}{8} r^4\)
Quarter circle with base on x-axis and side on y-axis. The x' axis intersects the centroid.
\(\left (\dfrac{4r}{3\pi}, \dfrac{4r}{3\pi} \right)\)
\(\bar{I}_{x'} = \dfrac{1}{2}\left(\dfrac{\pi}{8} - \dfrac{8}{9\pi}\right) r^4 \)
\(\bar{I}_{x'} \approx 0.0549\ r^4\)
\(\bar{I}_{y'} = \dfrac{\pi}{8} r^4 \)
\(I_x = I_y = \dfrac{\pi }{16}r^4\)
See Example 7.7.14 for proof. \(\dfrac{4 r}{3 \pi} \approx 0.424\ r\)