Trapezoid Section Properties Calculator
Enter the top width, bottom width and height of an isosceles trapezoid and get its area, centroid, second moments, section moduli for both fibres and radii of gyration, drawn to scale.
Section dimensions
Enter the dimensions in the selected unit — the drawing and the results update as you type. The drawing uses centroidal coordinates: the bold gridlines are the x and y axes through the centroid C.
Section
- Section with its dimensions
- Centroid C, centroidal axes x and y
- Principal axes 1 and 2, angle θp
- Shear centre S
Trapezoidal sections are common wherever a part tapers: dovetail slides, machine-tool beds, V-belt profiles, concrete beams and retaining walls, the flanges of tapered-flange channels. Like the triangle and the semicircle, the trapezoid has its centroid off-centre — closer to the wider side — so the section moduli at the two fibres differ.
With a = b the calculator reproduces the rectangle, with a = 0 the isosceles triangle; the formulas below cover the whole range. The shape is symmetric about the vertical axis, so Ixy = 0 and the centroidal x and y axes are principal.
How the trapezoid properties are calculated
- Area
- A = (a + b)·h / 2
- Centroid above the bottom edge
- ȳ = h·(2·a + b) / (3·(a + b))
- Second moment about the centroidal x axis
- Ix = h³·(a² + 4·a·b + b²) / (36·(a + b))
- Second moment about the centroidal y axis
- Iy = h·(a + b)·(a² + b²) / 48
- Elastic section moduli about x
- Sx,top = Ix / (h − ȳ), Sx,bot = Ix / ȳ
Principal moments and axes follow from Ix, Iy and Ixy with I1,2 = (Ix + Iy)/2 ± √[((Ix − Iy)/2)² + Ixy²] and tan 2θp = −2·Ixy/(Ix − Iy); the radii of gyration are r = √(I/A) and the elastic section moduli S = I/c for each extreme fibre. Definitions of every property are in the glossary on the section properties overview.
Assumptions. Isosceles trapezoid: the two parallel sides are horizontal and centred on each other. A skewed trapezoid has a product of inertia and rotated principal axes, which this page does not cover. No torsion constant is given.
Other sections
Frequently asked questions
Where is the centroid of a trapezoid?
At ȳ = h·(2a + b)/(3(a + b)) above the wider side b. Check the limits: a = b gives h/2 (rectangle), a = 0 gives h/3 (triangle). The calculator draws the ȳ dimension on the right of the section so the position is visible at a glance.
Can I enter a trapezoid that is wider at the top?
Yes — simply make a larger than b. The formulas are symmetric in the sense that the section is just flipped; the reported ȳ is still measured from the bottom edge, so it will then be larger than h/2.
Which section modulus should I use?
The smaller one, unless you know which fibre is critical. The fibre farther from the centroid — the narrower side — has the smaller S and therefore the higher bending stress for a given moment. For a ductile material in pure bending, the smaller S governs.
References & further reading
- List of centroids — Wikipedia — centroid positions of the elementary plane shapes.
- List of second moments of area — Wikipedia — closed-form Ix and Iy for the common shapes, used to check this calculator.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
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