Polar Moment of Inertia Calculator

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Cross-section type Solid circle Hollow circle (tube)

Outer diameter D

Inner diameter d (hollow only)

Results

Polar Moment of Inertia (J)

613,592.32

length⁴ (e.g. mm⁴ if D is in mm)

Polar section modulus Zp = J / (D/2)	24,543.69 length³

What is the polar moment of inertia?

The polar moment of inertia (J), also called the second polar moment of area, measures a cross-section's resistance to torsion (twisting). For circular shafts it appears directly in the torsion formula \(\tau = T\cdot r / J\), where T is the applied torque and r is the radial distance from the center. A larger J means the shaft twists less under the same torque. This calculator works for solid round bars and hollow round tubes.

The polar moment of inertia measures a circular shaft's resistance to twisting (torsion).

How to use this calculator

Choose whether your cross-section is a solid circle or a hollow circle (tube). Enter the outer diameter D. For a hollow section, also enter the inner (bore) diameter d. The result J is returned in length⁴ — so if you enter diameters in millimeters, J is in mm⁴; if in inches, J is in in⁴. The calculator also reports the polar section modulus \(Z_p = J/(D/2)\), useful for computing maximum surface shear stress as \(\tau = T/Z_p\).

The formula explained

For a solid circle of diameter D: $$J = \frac{\pi \text{D}^{4}}{32}$$ For a hollow circle, you subtract the bore's contribution: $$J = \frac{\pi(\text{D}^{4} - \text{d}^{4})}{32}$$ Because the diameter is raised to the fourth power, increasing diameter has a dramatic effect on torsional stiffness, while material near the center contributes very little — which is why hollow shafts are an efficient use of material.

Solid section (left) uses outer diameter D; hollow section (right) subtracts the inner diameter d.

Worked example

A hollow shaft has D = 50 mm and d = 30 mm. $$J = \frac{\pi(50^{4} - 30^{4})}{32} = \frac{\pi(6{,}250{,}000 - 810{,}000)}{32} = \frac{\pi \times 5{,}440{,}000}{32} = \pi \times 170{,}000 \approx 534{,}070.75 \text{ mm}^4$$ The section modulus \(Z_p = J/(25) \approx 21{,}362.83 \text{ mm}^3\).

FAQ

What's the difference between J and the area moment of inertia I? For a circle, \(J = 2I\) because \(J = I_x + I_y\) and \(I_x = I_y\). J governs torsion; I governs bending.

Can I use radius instead of diameter? This tool expects diameters. If you have radius, multiply by 2 first, or note that \(J = \pi r^{4}/2\) for a solid circle.

What units does J use? Whatever length unit you enter to the fourth power — keep diameters in consistent units (all mm or all inches).