Polar Moment of Inertia Calculator

The polar second moment of area, commonly written J, describes how cross-sectional area is distributed about an axis perpendicular to the section. It has dimensions of length to the fourth power. A larger value means more area lies farther from the polar axis, which generally improves a circular shaft's resistance to torsional stress and angular twist. This calculator accepts dimensions in metres and returns J in metres to the fourth power. For a solid circle of diameter d, J = πd⁴ / 32. For a hollow circle with outer diameter D and inner diameter d, J = π(D⁴ - d⁴) / 32. The fourth-power dependence explains why moving material outward is so effective: a hollow shaft can retain substantial torsional performance while using less material than a solid shaft of similar outside diameter. The inner diameter must remain smaller than the outer diameter, and all dimensions must use the same unit. For any planar area, the polar area moment about a point equals the sum of the two perpendicular area moments through that point: J = Ix + Iy. For a rectangle centered on the axis, this gives J = bh(b² + h²) / 12. However, this rectangular polar area moment is not generally the same as the Saint-Venant torsional constant used to calculate twist and torsional shear in a noncircular bar. The distinction matters because noncircular sections warp under torsion. The calculator labels the rectangular result as a polar area moment and should not be substituted blindly into a shaft-twist formula. For circular shafts, elastic torsion relationships include maximum shear stress τmax = Tr / J and angle of twist θ = TL / (GJ), where T is torque, r is outside radius, L is shaft length, and G is shear modulus. These relations assume homogeneous linear-elastic material, small deformation, uniform circular geometry, and loading about the centroidal longitudinal axis. Keyways, splines, cracks, abrupt diameter changes, and stress concentrations require additional analysis. Choose dimensions from the actual load-carrying section and preserve unit consistency. If dimensions are entered in millimetres, the numerical result is in millimetres to the fourth power rather than metres to the fourth power; convert before comparing with an SI calculation. Use this result for coursework, preliminary sizing, and formula checks. Detailed machine design should also evaluate strength, fatigue, deflection, manufacturing tolerances, stress concentrations, and the correct torsion theory for the chosen shape.