Polar Moment of Inertia Calculator

What Does the Polar Moment of Inertia Calculator Compute?

The Polar Moment of Inertia Calculator finds the second moment of area about an axis perpendicular to the cross-section plane (the polar moment Jp), as well as the second moments of area about the x-axis (Jx) and y-axis (Jy) individually, for five common structural cross-sections: solid circle, hollow circle (tube), solid rectangle, hollow rectangle (box section), and I-beam. These values are the primary inputs to torsional stress analysis, beam bending calculations, and shaft design. The polar moment of area Jp is the key quantity in the torsion formula: maximum shear stress = T times r divided by Jp, where T is the applied torque and r is the outermost radius. For rectangular and I-sections, the perpendicular axis theorem Jp = Jx + Jy applies. According to the Engineering Toolbox second moment of area reference, the polar moment of area is sometimes called the polar second moment of area or torsional constant (though strictly, the torsional constant K equals Jp only for circular cross-sections; for non-circular sections K is less than Jp due to warping). This calculator uses the exact second moment of area formulas for all five shapes.

All five cross-section shapes support four units (mm, cm, m, in) and display the polar moment, both individual second moments, and the cross-sectional area simultaneously. The formula reference table at the bottom shows the exact expression used for each shape so you can verify results manually or adapt the formula for partial sections.

The Core Formulas and Their Derivations

The second moment of area about the x-axis is Jx = integral over the cross-section of y squared dA. For a solid circle of radius r: Jx = Jy = pi times r to the fourth divided by 4, giving Jp = pi times r to the fourth divided by 2. For a hollow circle with outer radius ro and inner radius ri: Jp = pi times (ro to the fourth minus ri to the fourth) divided by 2. For a rectangle of width b and height h: Jx = b times h cubed divided by 12 (about the centroidal x-axis); Jy = h times b cubed divided by 12. The perpendicular axis theorem gives Jp = Jx + Jy for any planar section. For an I-beam with flange width bf, flange thickness tf, total depth d, and web thickness tw: Jx = (bf times d cubed minus (bf minus tw) times hw cubed) divided by 12, where hw = d minus 2 times tf is the web height. These are all centroidal second moments; for off-centroid axes, use the parallel axis theorem: J = J_centroid plus A times distance squared.

Torsional Stress and Shaft Design: Why Jp Matters

In a circular shaft under pure torque T, the shear stress at any radius r from the centreline is tau = T times r divided by Jp. The maximum shear stress occurs at the outer surface: tau_max = T times r_outer divided by Jp. For a solid steel shaft of radius 25 mm (r = 0.025 m) and Jp = pi times 0.025 to the fourth divided by 2 = 614,000 mm to the fourth = 6.14 times 10 to the minus 7 m to the fourth, carrying a torque of 500 N·m: tau_max = 500 times 0.025 divided by 6.14 times 10 to the minus 7 = 20.4 MPa. For the same torque on a hollow shaft (ro = 25 mm, ri = 20 mm): Jp = pi times (25 to the fourth minus 20 to the fourth) divided by 2 = 362,000 mm to the fourth. Maximum stress = 500 times 25 divided by 362,000 mm to the fourth = 34.5 MPa, higher per unit area but the hollow shaft uses 36 percent less material. According to the eFunda torsion formula reference, the torsional angle of twist phi = T times L divided by (G times Jp), where G is the shear modulus and L is the shaft length. Both the maximum stress and angle of twist are inversely proportional to Jp, confirming that maximising Jp minimises both failure risk and deformation for a given torque.

Our inclined plane calculator covers surface-friction statics, while polar moment of inertia is the foundation for the rotational counterpart: torque transmission through shafts and structural sections. For the mass-based rotational inertia (used in rotation dynamics), the relevant quantity is the mass moment of inertia I = integral of r squared dm, which this calculator does not compute.

Hollow vs Solid Sections: Material Efficiency

The hollow circle formula Jp = pi times (ro to the fourth minus ri to the fourth) / 2 shows that removing material from the centre of a shaft costs very little in torsional stiffness. For a solid shaft of radius 25 mm, Jp = 614,000 mm to the fourth. For a hollow shaft with the same outer radius but inner radius 20 mm (removing 64 percent of the area): Jp = 362,000 mm to the fourth, still 59 percent of the solid value despite using only 36 percent of the material. This is the mechanical efficiency argument for hollow shafts, tubes, and box sections: material near the neutral axis (centre) contributes proportionally to r squared to Jp, so it does far less work per unit mass than material at the outer radius. I-beams exploit the same principle: by concentrating material in the flanges (far from the neutral bending axis) and using a thin web, they achieve very high Jx values with minimal cross-sectional area.

The Steel Construction Institute structural design reference provides standardised Jx values for all UK and European standard I-section and hollow section steel profiles. This calculator lets you verify those published values from first principles by entering the actual measured dimensions of any section.

Accuracy and Limitations

This calculator uses the exact closed-form formulas for the second moments of area of uniform, homogeneous cross-sections. Results are accurate to full floating-point precision for the mathematical model. In practice, real sections differ from the ideal: I-beams have fillet radii at flange-web junctions that add area not captured by the simplified formula, and hollow sections have rounded corners. For standard structural steel sections, these corrections are already included in the published tabulated values from steel manufacturers and standards bodies; the simple formulas in this calculator underestimate Jx for I-beams by approximately 2 to 5 percent compared to tabulated values, depending on the section series. For preliminary design and educational calculations, the formulas here are entirely adequate. For final structural design, always use the published section properties from the relevant national steel design standard.

The torsional constant K for non-circular sections (rectangle, I-beam) is not the same as Jp. For a rectangle, K is substantially less than Jp and depends on the aspect ratio; for an I-beam, K is approximately the sum of the torsional constants of the individual plate elements. This calculator reports Jp (second moment of area) only. For St. Venant torsion analysis of non-circular sections, use published K values from structural steel tables rather than Jp from this calculator.

The Most Common Polar Moment of Inertia Mistake

The most common confusion in second moment of area calculations is mixing up the polar moment of area Jp (used in torsion, units of length to the fourth: mm to the fourth) with the mass moment of inertia I (used in rotational dynamics, units of mass times length squared: kg·m squared). Both are called "moment of inertia" in different contexts, which creates persistent naming confusion. The polar moment of area depends only on the shape and dimensions of the cross-section; it is a purely geometric property. The mass moment of inertia depends on both the geometry and the material density distribution. A steel shaft and an aluminium shaft of identical dimensions have identical Jp values but very different mass moments of inertia. The second most common mistake is applying the torsion formula tau = T times r divided by Jp to a non-circular section: this formula is exact only for circular sections. For rectangular sections under torsion, the stress distribution is not linear in radius, and the correct formulas involve correction factors. Always confirm which type of section you are analysing before applying the simple circular torsion formula.