Polar Moment Calculator - J for Solid and Hollow Cross-Sections

A polar moment calculator returns the second polar moment of area J for a cross-section from its shape and dimensions, then reports J in m^4, cm^4, mm^4, and in^4 from one entry. The result drops into shaft-torsion sizing where J is the section property the textbook formula needs first.

J is also called the polar moment of inertia in mechanics texts. For circular cross-sections J equals the torsional constant K used in tau = T*r/J and theta = T*L/(G*J); for non-circular sections J is only the geometric integral, so tau and GJ readouts are first-order estimates. J is not the area moment of inertia I_x or I_y, and the m^4, cm^4, mm^4, and in^4 readouts cover the units used in SI, European, and imperial shaft tables.

When the engineering problem actually needs the area moments I_x and I_y for bending stress, the Moment of Inertia Calculator gives the closed-form Ix and Iy for the same shapes, and the polar moment equals Ix plus Iy for any centroidal section.

The polar moment calculator reads the active cross-section and the dimension fields that belong to it, then applies the closed-form integral result for that shape. The same J value is converted to cm^4, mm^4, and in^4 from the m^4 result, and the optional G and T fields drive the GJ stiffness and tau readouts.

• J: Polar moment in m^4 (also reported in cm^4, mm^4, in^4).
• d: Outer diameter of a solid circle in metres.
• d_o, d_i: Outer and inner diameters of a hollow circle in metres.
• a, b: Width and height of a rectangle in metres.
• R_m, t: Mean radius and wall thickness of a thin-walled tube in metres.
• G: Shear modulus in GPa.
• T: Applied torque in N*m.

For a solid circular cross-section the integral J = integral r^2 dA over the disc area reduces to pi * d^4 / 32. The hollow tube subtracts the inner disc to give pi * (d_o^4 - d_i^4) / 32, the rectangle integrates r^2 = x^2 + y^2 to give a*b*(a^2 + b^2)/12, and the thin-walled tube form 2*pi*R_m^3*t is the leading-order term. Once J is in hand, the calculator multiplies it by G to expose the GJ torsional rigidity, the term in the denominator of theta = T*L/(G*J), and the tau readout uses the outer radius so tau_max = T*r/J matches the textbook expression.

According to Wikipedia - Polar moment of inertia, the polar moment of inertia of a solid circular cross-section of diameter d is J = pi * d^4 / 32 and the hollow circular form is J = pi * (d_o^4 - d_i^4) / 32

The next step after a polar moment is the angle of twist, and the Angle of Twist Calculator takes J from this page and combines it with torque, length, and shear modulus to return theta in radians and degrees.

Four ideas sit underneath every polar moment calculator: the polar integral J = integral r^2 dA, the link J = I_x + I_y, the torsion link theta = T*L/(G*J), and the thin-wall approximation J = 2*pi*R_m^3*t.

The J = I_x + I_y relationship makes a polar moment a good companion to a bending-stress check, so the Beam Bending Stress Calculator uses the same rectangular section data to return the bending stress sigma = M*c / I.

Pick the cross-section, fill in the dimensions that belong to that shape, and read J in the unit you need. Add T and G for tau and GJ.

1. 1 Pick the shape: Use the shape selector to switch between Solid Circle, Hollow Circle, Solid Rectangle, Hollow Rectangle, and Thin-Walled Tube so the dimension row matches the section.
2. 2 Enter the active dimensions: For Solid Circle enter d; for Hollow Circle enter d_o and d_i with d_i less than d_o; for a rectangle enter a and b; for a thin-walled tube enter R_m and t with t much smaller than R_m.
3. 3 Read J in the unit you need: The primary card shows J in m^4; the other rows give the same value in cm^4, mm^4, and in^4.
4. 4 Add torque and shear modulus: Enter T in N*m and G in GPa to get tau_max = T*r/J on the outer surface and GJ = G*J for the angle of twist formula.

For a 50 mm solid steel drive shaft carrying 100 N*m, pick Solid Circle, enter d = 0.05, T = 100, and G = 79.3. The calculator returns J = 6.14e-7 m^4, tau = 4.07 MPa, and GJ = 4.87e4 N*m^2.

When the design is being driven by a known power and a known shaft speed, the Torque to Horsepower Calculator converts horsepower to torque in N*m so the same torque value can be fed into the tau and GJ readouts on this page.

The polar moment calculator keeps the shape-specific J formulas, the unit conversions, and the torsion outputs in one panel, so J feeds angle of twist, torsional stiffness, and shear stress checks without manual unit work.

• • Five shapes in one panel: Switch between Solid Circle, Hollow Circle, Solid Rectangle, Hollow Rectangle, and Thin-Walled Tube on the same page.
• • m^4, cm^4, mm^4, and in^4 together: J is reported in all four common area-moment units from one calculation, so the result pastes into SI, European, and imperial shaft tables.
• • Tau and GJ from the same J: Adding T and G exposes tau_max = T*r/J and GJ = G*J in the same panel.

Because the polar moment is a pure geometric integral, the result does not depend on the material, and the same J with a different G drives the entire torsion check.

For a shaft that also carries transverse loads, the same cross-section needs a shear and bending diagram, and the Shear Force Bending Moment Calculator maps a beam loading into the V and M curves that combine with J for a combined-loading check.

The polar moment of inertia scales with a power of every dimension, so the four cards below show how each input changes J, the GJ stiffness, and the maximum shear stress.

• • Reports the second polar moment of area J in m^4 (or cm^4, mm^4, in^4). It does not compute the mass polar moment I = integral r^2 dm in kg*m^2 used by rigid-body rotation.
• • The thin-walled tube formula 2*pi*R_m^3*t is a leading-order approximation. Once t exceeds roughly R_m/5, switch to the exact hollow-circle formula.
• • tau_max and GJ use the torsional constant assumption: exact for solid and hollow circular cross-sections (J equals K) and a first-order estimate for rectangular and thin-walled sections. Check with a torsional-constant solver before final sizing.

For practical shaft sizing, the next step after the polar moment is to combine it with the length, the torque, and the shear modulus to get the angle of twist. The GJ readout is the torsional rigidity in the denominator of theta = T*L/(G*J), so the next step is to multiply by length L and applied torque T, then divide by GJ.

According to Engineering Toolbox - Torsion of Shafts, the angle of twist of a circular shaft is theta = T * L / (G * J) with the polar moment of inertia J of a solid circular cross-section equal to pi * d^4 / 32, and structural steel has a shear modulus around 79.3 GPa

According to Wikipedia - Torsion (mechanics), the angle of twist of a circular shaft is theta = T * L / (G * J) and the maximum elastic shear stress is tau_max = T * r / J, where J is the polar moment of inertia of the cross-section

The tau_max readout on this page assumes a smooth shaft, so once a keyway, shoulder, or fillet is added the Stress Concentration Factor Calculator returns the Kt multiplier that lifts the local stress above the polar-moment-only estimate.