Polar Moment of Inertia Calculator
Compute the polar moment of inertia J for solid circles, hollow circular tubes, and rectangular cross-sections. J determines a shaft's resistance to twisting and is foundational in torsion design.
Cross-section shape
mm
Second polar moment of area about the centroid
- 1
Radius R = D ÷ 2
100 ÷ 2 = 50 mm - 2
R⁴
50⁴ = 6,250,000 mm⁴ - 3
J = π × R⁴ ÷ 2
π × 6,250,000 ÷ 2 = 9,817,477.04 mm⁴
How does this calculator work?
The polar moment of inertia J = Ix + Iy resists torsion. For a solid circle J = πR⁴/2; for a hollow tube J = π(R⁴ − r⁴)/2; for a rectangle J = bh(b² + h²)/12. Shear stress τ = T·c/J and twist angle φ = TL/(GJ), where T is torque, c is outer radius, G is shear modulus, and L is shaft length.
Formula
How this is calculated
The polar moment of inertia J (also called the second polar moment of area) measures how a cross-section resists torsional (twisting) loads. It is the sum of the two second moments of area about the centroidal axes: J = Ix + Iy. A larger J means the section is stiffer in torsion and the shear stress for a given torque is lower.
For a solid circular shaft of outer radius R, the formula is J = πR⁴/2 (equivalently πD⁴/32 in terms of diameter D). For a hollow tube with outer radius R and inner radius r, the hollow portion is subtracted: J = π(R⁴ − r⁴)/2. Circles are the most efficient shape for torsion because all material is equidistant from the centre. A rectangle uses J = Ix + Iy = bh³/12 + hb³/12 = bh(b² + h²)/12, but rectangles are significantly less efficient in torsion than circular sections of the same area.
This calculator gives J in mm⁴ when dimensions are entered in millimetres. Scale by the fourth power for other units (1 m⁴ = 10¹² mm⁴). The torsion formula τ = T·c/J relates shear stress τ at radius c to applied torque T; angle of twist φ = TL/(GJ) where G is the shear modulus and L is shaft length.
Frequently asked questions
The second moment of area (Ix or Iy) is taken about a single planar axis and governs bending stiffness. The polar moment of inertia J is taken about the axis perpendicular to the cross-section (z-axis) and equals Ix + Iy by the perpendicular axis theorem. J governs torsional stiffness.
Material near the centre of a solid shaft contributes little to J (because it is close to the twist axis) but adds weight. Removing the core with a hollow shaft gives almost the same J at much lower mass, which is why hollow drive shafts and tubes are common in engineering.
J has units of length⁴. If dimensions are in mm, J is in mm⁴. To convert: 1 in⁴ = 416 231 mm⁴; 1 m⁴ = 10¹² mm⁴. Always confirm the units match those expected by your torsion or finite-element software.
Also known as
TG we-Calculate Editorial Team. (2026). Polar Moment of Inertia Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polar-moment-calculator
TG we-Calculate Editorial Team. "Polar Moment of Inertia Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/polar-moment-calculator.
TG we-Calculate Editorial Team, "Polar Moment of Inertia Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polar-moment-calculator
@misc{wecalculate_polar_moment_calculator, title = {Polar Moment of Inertia Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polar-moment-calculator}}, year = {2026}, note = {TG we-Calculate} }
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