Moment of Inertia and Centroid of a Half Circle

Section properties of a Half Circle: area, centroid location and second moment of area (moment of inertia) about the centroidal axes and about the base and left edge. These values are used to size beams and columns, compute bending stresses (σ = M·y/I) and deflections.

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Half Circle cross section with its dimensions and centroid

$r$ = radius

Property	Symbol	Formula
Area	$A$	$\frac{\pi r^2}{2}$
Perimeter	$P$	$\pi r + 2r$
Centroid from the left edge	$\bar{x}$	$0$
Centroid from the bottom edge	$\bar{y}$	$\frac{4r}{3\pi}$
Moment of inertia about the centroidal x-axis	$\bar{I}_x$	$\left(\frac{\pi}{8} - \frac{8}{9\pi}\right) r^4$
Moment of inertia about the centroidal y-axis	$\bar{I}_y$	$\frac{\pi r^4}{8}$
Moment of inertia about the base	$I_x$	$\frac{\pi r^4}{8}$
Moment of inertia about the left edge	$I_y$	$\bar{I}_y$

The origin is the bottom-left corner of the bounding box. The moments of inertia about the base and the left edge follow from the parallel axis theorem: $I_x = \bar{I}_x + A\bar{y}^2$ .

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Moment of Inertia and Centroid of a Half Circle

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