Moment of Inertia and Centroid of a Rotated Rectangle

Section properties of a Rotated Rectangle: 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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Rotated Rectangle cross section with its dimensions and centroid
  • bb = base
  • hh = height
  • θ\theta = rotation angle
PropertySymbolFormula
AreaAAbhb h
PerimeterPP2(b+h)2(b + h)
Centroid from the left edgexˉ\bar{x}bcos⁡(θ)+hsin⁡(θ)2\frac{b \cos(\theta) + h \sin(\theta)}{2}
Centroid from the bottom edgeyˉ\bar{y}bsin⁡(θ)+hcos⁡(θ)2\frac{b \sin(\theta) + h \cos(\theta)}{2}
Moment of inertia about the centroidal x-axisIˉx\bar{I}_xbh12(b2sin⁡2(θ)+h2cos⁡2(θ))\frac{b h}{12} (b^2 \sin^2(\theta) + h^2 \cos^2(\theta))
Moment of inertia about the centroidal y-axisIˉy\bar{I}_ybh12(b2cos⁡2(θ)+h2sin⁡2(θ))\frac{b h}{12} (b^2 \cos^2(\theta) + h^2 \sin^2(\theta))
Moment of inertia about the baseIxI_xIˉx+yˉ2A\bar{I}_x+\bar{y}^2 A
Moment of inertia about the left edgeIyI_yIˉy+xˉ2A\bar{I}_y+\bar{x}^2 A

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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