Moment of Inertia and Centroid of a Circle Sector

Section properties of a Circle Sector: 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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Circle Sector cross section with its dimensions and centroid
  • rr = radius
  • θ\theta = sector angle
PropertySymbolFormula
AreaAAθr22\frac{\theta r^2}{2}
PerimeterPPθr+2r\theta r + 2r
Centroid from the left edgexˉ\bar{x}4rsin⁡(θ/2)3θ\frac{4r \sin(\theta/2)}{3\theta}
Centroid from the bottom edgeyˉ\bar{y}00
Moment of inertia about the centroidal x-axisIˉx\bar{I}_xr4(θ−sin⁡θ)8\frac{r^4(\theta - \sin\theta)}{8}
Moment of inertia about the centroidal y-axisIˉy\bar{I}_yIy−xˉ2AI_y - \bar{x}^2 A
Moment of inertia about the baseIxI_xIˉx\bar{I}_x
Moment of inertia about the left edgeIyI_yr4(θ+sin⁡θ)8\frac{r^4(\theta + \sin\theta)}{8}

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: Ix=Iˉx+Ayˉ2I_x = \bar{I}_x + A\bar{y}^2. The angle θ is entered in degrees in the calculator; in the formulas it is in radians.

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Enter the dimensions and get the area, centroid and moments of inertia instantly, with the section drawn to scale. Metric and imperial units.

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