Torsional Constant Calculator — Polar Moment of Inertia
Calculate the torsional constant J (polar second moment of area) for solid or hollow circular cross-sections. J is the key geometric property in torsion: it appears in the shear stress formula τ = Tr/J and the angle-of-twist formula φ = TL/(GJ).
Cross-section type
mm
Polar second moment of area — the resistance of the cross-section to twisting
- 1
d⁴
50⁴ = 6,250,000 - 2
Torsional constant J = π × d⁴ ÷ 32
π × 6,250,000 ÷ 32 = 613,592.32
How does this calculator work?
The torsional constant J = πd⁴/32 (solid) or π(dₒ⁴−dᵢ⁴)/32 (hollow) quantifies a circular cross-section's resistance to twisting. Enter the diameter to get J in mm⁴, then use τ = Tr/J for shear stress and φ = TL/(GJ) for angle of twist.
Formula
How this is calculated
When a shaft is twisted by an applied torque T, the shear stress at any point and the total angle of twist both depend on how the cross-sectional area is distributed relative to the neutral axis. This geometric property is the torsional constant J, also called the polar second moment of area (or polar moment of inertia for a thin-walled section). For a solid circular shaft of diameter d the formula is J = πd⁴/32. For a hollow shaft (tube) with outer diameter dₒ and inner diameter dᵢ, the formula is J = π(dₒ⁴ − dᵢ⁴)/32 — the hollow region simply subtracts its polar moment from the solid value.
With J known you can compute: maximum shear stress τ_max = T·r/J (where r is the outer radius and T is the applied torque); and angle of twist φ = T·L/(G·J) (where L is shaft length and G is the shear modulus of the material). The calculator also outputs the bending second moment of area I = J/2, which applies when the same circular section is loaded in bending rather than torsion.
Note: J = πd⁴/32 strictly applies only to solid or hollow circular cross-sections. For rectangular, I-section or thin-walled open sections, J must be computed differently (often approximated as (1/3)Σbt³ for thin rectangles). The area moment of inertia I for bending is also only J/2 for circular sections.
Frequently asked questions
I (second moment of area or moment of inertia) resists bending about one axis; J (polar second moment of area or torsional constant) resists twisting about the longitudinal axis. For a circular cross-section J = 2I because the polar moment is the sum of the two orthogonal bending moments.
J scales with d⁴. Doubling the diameter increases J by a factor of 16. This is why relatively small increases in shaft diameter drastically reduce shear stress and angle of twist under the same torque.
Yes — and often stronger. Removing core material (where stress is lowest) reduces weight while preserving most of the torsional stiffness. A hollow shaft with the same mass as a solid shaft typically has a larger outer diameter and therefore a higher J, giving lower stress.
Also known as
TG we-Calculate Editorial Team. (2026). Torsional Constant Calculator — Polar Moment of Inertia [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torsional-constant-calculator
TG we-Calculate Editorial Team. "Torsional Constant Calculator — Polar Moment of Inertia." TG we-Calculate. 2026. https://we-calculate.com/calculator/torsional-constant-calculator.
TG we-Calculate Editorial Team, "Torsional Constant Calculator — Polar Moment of Inertia," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torsional-constant-calculator
@misc{wecalculate_torsional_constant_calculator, title = {Torsional Constant Calculator — Polar Moment of Inertia}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torsional-constant-calculator}}, year = {2026}, note = {TG we-Calculate} }
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