Moment of Inertia and Centroid of a Rectangle

Section properties of a 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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Rectangle cross section with its dimensions and centroid
  • bb = base
  • hh = height
PropertySymbolFormula
AreaAAbhb h
PerimeterPP2(b+h)2(b + h)
Centroid from the left edgexˉ\bar{x}b2\frac{b}{2}
Centroid from the bottom edgeyˉ\bar{y}h2\frac{h}{2}
Moment of inertia about the centroidal x-axisIˉx\bar{I}_xbh312\frac{b h^3}{12}
Moment of inertia about the centroidal y-axisIˉy\bar{I}_yb3h12\frac{b^3 h}{12}
Moment of inertia about the baseIxI_xbh33\frac{b h^3}{3}
Moment of inertia about the left edgeIyI_yb3h3\frac{b^3 h}{3}
Plastic section modulus about xZxZ_xbh24\frac{b h^2}{4}
Plastic section modulus about yZyZ_yb2h4\frac{b^2 h}{4}
Torsional constantJJ13bh3[1−0.63hb(1−h412b4)]\frac{1}{3}bh^3 \left[ 1-0.63\frac{h}{b} \left( 1-\frac{h^4}{12b^4} \right) \right]

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.

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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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