Torsional Constant Calculator

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Calculate torsional constant J for common cross-sections and quickly see when J matches polar moment and when it does not.

Select the cross-section whose torsional constant J you want to calculate. Use a standard shape with the matching dimensions.
All dimension inputs use this same length unit, and the output J will be shown in this unit to the fourth power.
Used for a solid round section. This is the full outside diameter.
Used for a hollow round section. This is the outside diameter.
Used for a hollow round section. This is the inside diameter and must be smaller than the outer diameter.
For a rectangle, use the shorter side here when possible. The calculator can still work if the sides are swapped, but formulas are usually written with b as the smaller side.
For a rectangle, this is the longer side when possible. For the thin open rectangle approximation, it is the centerline side length of the longer segment.
Used for the thin-walled open built-up approximation. Enter the centerline length of wall segment 1.
Used for the thin-walled open built-up approximation. Enter the wall thickness for segment 1.
Used for the thin-walled open built-up approximation. Enter the centerline length of wall segment 2.
Used for the thin-walled open built-up approximation. Enter the wall thickness for segment 2.
Used for the thin-walled open built-up approximation. Enter the centerline length of wall segment 3.
Used for the thin-walled open built-up approximation. Enter the wall thickness for segment 3.
Advanced options
Display
Choose how numbers are shown. Auto uses standard decimals for moderate values and scientific notation only for very small or very large results.
For circular sections, J equals the polar moment of inertia. For non-circular sections, this comparison helps show the difference when a simple polar-moment formula would be misleading.
Optional extra wall segments (thin-walled built-up)
Optional extra wall segment for the thin-walled open built-up approximation. Leave blank if not used.
Optional extra wall thickness for the thin-walled open built-up approximation. Leave blank if not used.
Optional extra wall segment for the thin-walled open built-up approximation. Leave blank if not used.
Optional extra wall thickness for the thin-walled open built-up approximation. Leave blank if not used.
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How to use our Torsional Constant Calculator

  1. Choose Cross-section shape that matches your part: solid round, hollow round, solid rectangle, thin-walled open rectangle approximation, or thin-walled open built-up section approximation.
  2. Select the Length unit first. Every dimension field uses that same unit, and the main result will be shown in that unit to the fourth power.
  3. Enter only the dimensions used by your selected shape. For circles, use Diameter (selected unit) or the pair Outer diameter (selected unit) and Inner diameter (selected unit).
  4. For a solid rectangle, enter Width b (selected unit) and Height h (selected unit). If you swap them, the calculator will reorder them internally so the result stays the same.
  5. For thin-walled open built-up sections, enter complete length-thickness pairs such as Segment 1 length (selected unit) and Segment 1 thickness (selected unit). Add optional segments only as full pairs.
  6. Open Advanced options if you want a different Display format, want to turn Show polar moment comparison on or off, or need optional extra thin-wall segments.
  7. Click Calculate. Review Torsional constant J, Torsional constant J in m^4, and Method note before copying the result into another formula.
  8. Sanity-check the output: J should be positive, circular sections should show a matching Polar moment comparison, and changing a dimension by a small amount should not create a wildly wrong result unless that dimension is raised to the fourth power.

Definitions

Cross-section shape: The outline of the part after you cut it and look straight at the end.

Torsional constant J: A geometric property used in Saint-Venant torsion formulas. It tells how the shape resists twisting, and its units are length^4 [4].

Polar moment comparison: A teaching comparison value. For circular sections it equals J, but for non-circular sections it generally does not [4][4].

Width b (selected unit): One side of a rectangle. In the rectangle formula, b is usually the shorter side.

Height h (selected unit): The other side of a rectangle. In the rectangle formula, h is usually the longer side.

Segment length and thickness: For thin-walled open sections, length is measured along the wall centerline and thickness is the wall thickness used in the approximation.

Torsional constant J in m^4: The same result converted to SI base units, useful when the rest of your equation uses meters.

Method note: A short message that tells you whether the result came from a standard exact circular formula or from an approximation.

Common mistakes and quick fixes

Mistake: Entering an inside size larger than the outside size for Inner diameter (selected unit) and Outer diameter (selected unit) .
Fix: Make sure Inner diameter (selected unit) is smaller than Outer diameter (selected unit) before calculating.

Mistake: Mixing units after choosing Length unit , such as typing inches into one field and millimeters into another.
Fix: Use one consistent unit for every dimension field, or change Length unit first and then re-enter all dimensions in that unit.

Mistake: Using the solid rectangle formula for a thin sheet-like open section by filling only Width b (selected unit) and Height h (selected unit) .
Fix: If the section is thin-walled and open, switch Cross-section shape to a thin-wall option and enter segment lengths and thicknesses instead.

Mistake: Entering a thickness that is as large as or larger than a wall length, such as Segment 1 thickness (selected unit) greater than Segment 1 length (selected unit) .
Fix: For thin-walled approximations, each thickness must be positive and smaller than its matching segment length.

Mistake: Filling Segment 4 length (selected unit) but leaving Segment 4 thickness (selected unit) blank, or the other way around.
Fix: Optional segments must be entered as complete pairs. Either fill both fields for that segment or leave both blank.

Mistake: Treating Polar moment comparison as if it were always the same property as Torsional constant J .
Fix: For round sections they match, but for non-circular sections use Torsional constant J when your torsion formula specifically asks for J.

Limitations & Key Assumptions / Boundary Conditions

  • For Solid rectangle, the calculator uses a standard engineering approximation, not an exact closed-form solution for all cases.
  • For Thin-walled open rectangle approximation and Thin-walled open built-up section approximation, the result is valid only when walls are thin compared with their segment lengths.
  • Thin-walled open-section formulas are not for closed-cell torsion. A closed box section behaves differently and usually has much higher torsional stiffness.
  • The thin-wall inputs use centerline segment lengths. Using outside or inside wall lengths instead will change the result.
  • For non-circular sections, the Polar moment comparison is only a comparison aid. Do not replace Torsional constant J with polar moment in torsion formulas unless your method specifically allows it.
  • All dimensions must use one consistent Length unit. Since J scales with the fourth power of length, even a small unit mistake creates a very large output error.
  • This calculator reports section property only. It does not calculate angle of twist, stress, torque capacity, warping effects, or material behavior.

Methodology

What the calculator computes

The calculator returns the Saint-Venant torsional constant J for the selected cross-section. It also converts that result to m^4 and shows a short method note so you know whether the value came from an exact circular formula or an approximation [4].

Shape formulas used

Solid round: J = (π/32) d^4

Here d is Diameter (selected unit). For a solid circular section, J equals the polar moment of inertia of area.

Hollow round: J = (π/32) (D^4 - d^4)

Here D is Outer diameter (selected unit) and d is Inner diameter (selected unit). For a circular tube, J also equals the polar moment.

Solid rectangle: J = a b^3 (1/3 - 0.21(b/a)(1 - b^4/(12a^4)))

Before using this formula, the calculator internally sorts the two sides so a is the longer side and b is the shorter side. That is why swapping Width b (selected unit) and Height h (selected unit) gives the same result. This is a standard approximation for rectangular sections [3][3].

Thin-walled open built-up section: J = sum((1/3) b_i t_i^3)

Each wall segment contributes one term using its centerline length b_i and thickness t_i. The calculator sums the filled segment pairs.

Thin-walled open rectangle approximation: J = (1/3)(2 h t^3 + 2 b t^3)

This is a special thin-walled open-section case with uniform thickness t. The calculator uses the same thin-wall idea: each wall segment contributes in proportion to length times thickness cubed.

SI conversion: J_m4 = J_input x (length unit in meters)^4

If you pick inches, the conversion factor is 0.0254 m per in, so the SI result is J x 0.0254^4.

Polar moment comparison

For circular sections, the comparison value matches J. For non-circular sections, the calculator may show a comparison note instead of suggesting they are interchangeable. This matters because students often confuse torsional constant with polar moment, but they are generally different outside circular sections [2][2].

Rectangle comparison only: I_p = (b h^3 + h b^3)/12

This rectangle expression is shown only as a learning comparison, not as the J formula for a solid rectangle.

Validation and edge handling

All used dimensions must be greater than zero. For hollow round sections, inner diameter must be smaller than outer diameter. For thin-walled open approximations, each thickness must be positive and smaller than its matching segment length. Optional segments must be entered as full length-thickness pairs. Hidden fields are ignored completely so they do not affect the active shape.

Worked mini-example

Suppose you choose Solid round, set Length unit to inches, and enter Diameter (selected unit) = 2.

J = (π/32) x 2^4 = (π/32) x 16 = π/2 = 1.5708 in^4

Then convert to SI:

J_m4 = 1.5708 x 0.0254^4 = 6.5373 x 10^-7 m^4

If you switched to a rectangular section with the same rough size, J would not generally equal the polar moment comparison value, which is exactly why this calculator keeps them separate.

Assumptions behind the result

Circular formulas are exact for the listed dimensions. The rectangle result is an engineering approximation, and thin-walled open-section results assume thin walls, open geometry, and centerline wall lengths. Real parts can differ if fillets, corner radii, thick walls, closed cells, or warping effects are important.

Sources

  1. Torsional Modulus Constant | Oasys GSA Documentation - Oasys-software
  2. Torsion Constant of a Rectangle - sectionproperties 2.1.5 documentation - Readthedocs
  3. Torsional behaviour of high strength steel (S700) rectangular hollow section stub column | Scientific Reports - Nature
  4. What is the Torsion Constant? - Projectengineer