Moment of Inertia and Centroid of an Isosceles Trapezoid

Section properties of an Isosceles Trapezoid: 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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Isosceles Trapezoid cross section with its dimensions and centroid
  • aa = top base
  • bb = bottom base
  • hh = height
PropertySymbolFormula
AreaAAh(a+b)2\frac{h(a + b)}{2}
PerimeterPPa+b+4h2+(a−b)2a + b + \sqrt{4h^2 + \left( a-b \right)^2}
Centroid from the left edgexˉ\bar{x}b2\frac{b}{2}
Centroid from the bottom edgeyˉ\bar{y}h(2a+b)3(a+b)\frac{h(2a + b)}{3(a + b)}
Moment of inertia about the centroidal x-axisIˉx\bar{I}_xh3(a2+4ab+b2)36(a+b)\frac{h^3(a^2 + 4ab + b^2)}{36(a + b)}
Moment of inertia about the centroidal y-axisIˉy\bar{I}_yh(a+b)(a2+b2)48\frac{h(a + b)(a^2 + b^2)}{48}
Moment of inertia about the baseIxI_x(3a+b)h312\frac{(3a + b)h^3}{12}
Moment of inertia about the left edgeIyI_yh(a+b)(a2+7b2)48\frac{h(a + b)(a^2 + 7b^2)}{48}

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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