Hollow Circle (Round Tube) Section Properties Calculator
Enter the outer diameter and wall thickness of a round tube or pipe and get its area, second moments, polar moment, section moduli and radius of gyration, with the cross-section 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
A round tube keeps most of the bending and torsional stiffness of the solid bar while removing the material near the centre that contributes least — the second moment grows with the fourth power of the distance from the axis, so the core of a solid shaft carries very little. That is why hollow sections dominate wherever weight matters: bicycle frames, hydraulic cylinders, space frames, drive shafts.
The calculator takes the outer diameter D and the wall thickness t (the usual tube designation, e.g. 60.3 × 3.2) and works with the inner diameter d = D − 2t. As for the solid circle, the polar moment Ip is the exact torsion constant J, since the section stays circular.
How the hollow circle (tube) properties are calculated
- Inner diameter
- d = D − 2·t
- Area
- A = π·(D² − d²) / 4
- Second moment of area
- Ix = Iy = π·(D⁴ − d⁴) / 64
- Polar moment = torsion constant
- Ip = J = π·(D⁴ − d⁴) / 32
- Elastic section modulus
- S = π·(D⁴ − d⁴) / (32·D)
- Plastic section modulus
- Z = (D³ − d³) / 6
- Radius of gyration
- r = √(D² + d²) / 4
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. Exact for a concentric tube with uniform wall. Seam-welded or drawn tubes with an eccentric bore should be treated with the minimum wall thickness.
Other sections
Frequently asked questions
Is a thin-walled approximation good enough for a tube?
For thin walls the common shortcut is I ≈ π·r_m³·t with r_m the mean radius. At D/t = 20 it under-estimates I by about 0.25%, at D/t = 8 by about 1.5%. This calculator always uses the exact difference of fourth powers, so there is no need to decide.
How much stiffness do I lose by boring out a solid shaft?
Much less than the weight you save. Boring a hole of half the outer diameter removes 25% of the area but only 6.25% of I and J (0.5⁴ = 0.0625). The radius of gyration result shows this directly: it increases as the wall gets thinner.
Can I use this for a pipe specified by nominal size?
Yes, but enter the actual outside diameter and wall thickness from the pipe schedule table, not the nominal bore — for a DN50 / 2" schedule 40 pipe, for example, D = 60.3 mm and t = 3.91 mm.
References & further reading
- 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.
- Torsion constant — Wikipedia — why J differs from the polar moment for non-circular sections, with the thin-walled formulas.
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