Rectangle Section Properties Calculator
Enter the width and height of a rectangular bar, plate or beam and get its area, second moments, section moduli, radii of gyration and torsion constant, 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
The rectangle is the reference section of strength of materials: Ix = bh³/12 is the first formula every engineer learns, and the parallel-axis theorem turns every I-beam, channel and box into a sum of rectangles. The cubic dependence on height is the whole reason beams are tall and narrow — doubling the depth multiplies the bending stiffness by eight and the strength by four.
The calculator also reports the plastic section modulus Z = bh²/4 (1.5 times the elastic S = bh²/6, the “shape factor” of the rectangle) and Roark’s torsion constant, which for a flat plate tends to the thin-strip value J ≈ b·t³/3.
How the rectangle properties are calculated
- Area
- A = b·h
- Second moments of area
- Ix = b·h³ / 12, Iy = h·b³ / 12
- Elastic section moduli
- Sx = b·h² / 6, Sy = h·b² / 6
- Plastic section moduli
- Zx = b·h² / 4, Zy = h·b² / 4
- Radii of gyration
- rx = h / √12, ry = b / √12
- Torsion constant (Roark)
- J ≈ a·b³·[1/3 − 0.21·(b/a)·(1 − b⁴/12a⁴)], a = long side, b = short side
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. Sharp corners. The torsion constant is Roark’s approximation of the Saint-Venant solution (about 1% accuracy); for very thin strips it converges to a·b³/3.
Other sections
Frequently asked questions
Which axis is the “strong axis” of a rectangle?
The axis parallel to the short side, i.e. the one you bend about when the long side is vertical. With b = 50 and h = 100, I_x = 4.17·10⁶ mm⁴ but I_y is only 1.04·10⁶ mm⁴ — a factor of (h/b)² = 4. The calculator reports the principal angle as 0° when x is the strong axis and 90° when y is.
What is the shape factor of a rectangle?
Z/S = (bh²/4)/(bh²/6) = 1.5. It is the ratio between the fully plastic moment and the moment at first yield, and it is the highest among the common structural sections (I-beams are around 1.10–1.15) because so much of the rectangle’s material sits near the neutral axis, where it only starts working after yield.
Can I use bh³/12 for a plate in torsion?
No: bh³/12 is a bending property. For a flat plate of width a and thickness t in torsion, J ≈ a·t³/3 — four times smaller than the polar moment would suggest for a thin strip. The calculator’s torsion constant row gives Roark’s value, which is valid for any aspect ratio.
References & further reading
- Second moment of area — Wikipedia — definition, parallel-axis theorem and the sign convention for the product of inertia.
- Section modulus — Wikipedia — elastic and plastic section modulus, with a table of formulas by shape.
- Torsion constant — Wikipedia — why J differs from the polar moment for non-circular sections, with the thin-walled formulas.
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