Polar Moment of Inertia (J) Calculator

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Polar second moment of area J: A cross-section (area) property that measures how strongly a circular shape resists twisting. Units are length^4 (example: in^4 or mm^4).

Solid circular vs Hollow circular: Solid has no hole. Hollow has an outer size and an inner size (the hole).

Outer radius R (derived): The outside radius used in the math. If you enter diameter, R = (outer diameter)/2.

Inner radius r_i (derived): The hole radius used in the math. If you enter diameter, r_i = (inner diameter)/2. For solid sections, r_i = 0.

Applied torque T: The twisting load on the shaft (example: lbf*in or N*m).

Maximum shear stress tau_max: The largest torsional shear stress at the outside surface of a circular shaft, computed from T, R, and J.

Shear modulus G: A material stiffness number that links shear stress to shear strain in the elastic range [1][2]. Higher G means less twist for the same torque.

Angle of twist phi: The total twist over the shaft length, reported in radians (and also degrees).

Mistake: Choosing Dimension you know = Diameter but entering a radius value into Outer diameter (selected unit) .
Fix: Either switch to Radius and use Outer radius (selected unit) , or keep Diameter and enter the full diameter across the circle.

Mistake: For a hollow section, setting the inside size larger than (or equal to) the outside size.
Fix: Check Cross-section type and then recalculate. Make the inner diameter/radius strictly smaller than the outer diameter/radius. If the hole is 0, you can just use Solid circular.

Mistake: Mixing units when comparing results (for example, treating mm^4 as if it were in^4).
Fix: Match Length unit to your drawing, and use the converted outputs J (in^4) or J (mm^4) only for comparison in those exact units.

Mistake: Turning Torsion helper (optional) On but using the wrong torque unit (lbf*ft vs lbf*in, or N*m vs N*mm).
Fix: Re-check the unit that your torque is written in, then set Torque unit to match before trusting tau_max .

Mistake: Turning Compute angle of twist On but entering shear modulus in the wrong unit (for example typing 79.3 as if it were psi instead of GPa).
Fix: Confirm the magnitude and unit of G, then set Shear modulus unit to match (psi, ksi, Pa, MPa, or GPa) so phi is computed correctly.

Mistake: Using this J in a rotation/dynamics problem (mass moment of inertia) because it says "moment of inertia".
Fix: Check Cross-section type and then recalculate. Check units: this calculator outputs an area property in length^4 (like in^4 or mm^4). Mass moment of inertia uses mass*length^2 (like kg*m^2).

Circular-only torsion: The torsion helper formulas used here (stress and twist) are the clean circular-shaft relationships. For noncircular shapes (rectangles, I-beams, thin-walled open sections), torsion uses a different property (often called a torsion constant), so these results will not match well.

Geometry boundaries: The outer diameter/radius must be greater than 0. For Hollow circular, the inner diameter/radius must be 0 or more and strictly smaller than the outer size.

Area property, not mass inertia: J here is a geometric area property with units like in^4 or mm^4. It is not the mass moment of inertia used in dynamics or rotational kinetic energy.

Torsion helper assumptions: Stress and twist assume a straight, prismatic (same cross-section along the length) circular shaft in linear-elastic, small-deformation torsion. Features like keyways, splines, shoulders/fillets, short thick transitions, or stress concentrations can make real peak stress higher than tau_max.

Unit system consistency: For torsion results, torque and stress units must be in a consistent system (US customary or SI). If you mix systems, the number can be wrong even if it looks reasonable.

Conversions for J: When converting J between length units, the length conversion factor is raised to the 4th power. Small length unit mistakes can create very large J mistakes.

This calculator finds the polar second moment of area J for a circular cross-section (solid or hollow). Optionally, it uses J to compute torsion quantities for a circular shaft: maximum shear stress at the outside surface and the angle of twist.

1) Convert to radii If you choose Diameter, the calculator uses outer radius R = (outer diameter)/2 and inner radius r_i = (inner diameter)/2. If you choose Radius, it uses the radii you typed.

2) Compute J

Solid circular: J = (π/2)*R^4 = (π/32)*D^4

Hollow circular: J = (π/2)*(R^4 - r_i^4) = (π/32)*(D_o^4 - D_i^4)

If your sizes are in the selected length unit, then J is in (that unit)^4.

3) Optional torsion helper (circular shafts) If torque is provided, the maximum shear stress at the outer surface is:

τ_max = T*R/J

If twist is turned on and you also provide shaft length and shear modulus, the twist angle is:

φ = T*L/(J*G)

degrees = radians*(180/π)

Shear modulus G is a material property used in elastic torsion [1][2].

4) Unit conversions shown in results When J is converted between length units, the length conversion factor is raised to the 4th power:

J_to = J_from*(length_factor)^4

Mini example Solid circular, Diameter basis, length unit = in, outer diameter D = 2 in. Then R = 1 in and:

J = (π/2)*1^4 = 1.5707963268 in^4

With torque T = 1000 lbf*in, the outer-surface shear stress is:

τ_max = 1000*1/1.5707963268 = 636.6197724 lbf/in^2 (psi)

If L = 12 in and G = 11,500,000 psi, then:

φ = 1000*12/(1.5707963268*11500000) = 0.00066327384 rad = 0.0380 deg