Polar Moment of Inertia Calculator

Enter your cross-section type — solid circular or hollow circular — along with the outer diameter (and inner diameter for hollow sections) to calculate the polar moment of inertia (J). You also get the polar section modulus (Zp) and, optionally, the shear stress from an applied torque. Useful for shaft and torsion analysis in mechanical and structural engineering.

Select the shape of your cross-section.

mm

Outer (or full) diameter of the circular section.

mm

Inner diameter for hollow sections only. Leave 0 for solid.

N·mm

Enter a torque to compute shear stress at the outer fiber. Leave 0 to skip.

Results

Polar Moment of Inertia (J)

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Polar Section Modulus (Zp)

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Outer Radius (c)

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Max Shear Stress (τ)

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Section Area Breakdown

Frequently Asked Questions

What is the polar moment of inertia?

The polar moment of inertia (J) is a geometric property of a cross-section that measures its resistance to torsional (twisting) deformation. It is defined as the sum of the second moments of area about two perpendicular in-plane axes passing through the centroid — equivalent to the moment of inertia about an axis perpendicular to the plane of the cross-section.

Why do we need the polar moment of inertia?

The polar moment of inertia is critical for analyzing shafts and any member subjected to torsion. It appears directly in the torsion formula τ = Tρ/J, which relates applied torque (T), radial distance (ρ), and the resulting shear stress. Engineers use it to size drive shafts, turbine shafts, and other power-transmission components so they don't fail under twisting loads.

How do I calculate the polar moment of inertia of a solid circle?

For a solid circular cross-section of diameter D, the polar moment of inertia is J = πD⁴/32. Equivalently, using radius r, it is J = πr⁴/2. This formula applies to any solid round shaft such as a steel rod or drive axle.

How do I calculate the polar moment of inertia of a hollow cylinder?

For a hollow circular section with outer diameter D and inner diameter d, the formula is J = π(D⁴ − d⁴)/32. The inner void removes material and therefore reduces J compared to a solid section of the same outer diameter. Hollow shafts are often used to save weight while retaining reasonable torsional stiffness.

What are the units of polar moment of inertia?

The polar moment of inertia has units of length to the fourth power. In SI units this is typically mm⁴ or m⁴, and in imperial units it is in⁴. When using the torsion formula, be consistent — mix mm with N and you get MPa (N/mm²) for stress directly.

What is the difference between the area moment of inertia and the polar moment of inertia?

The area moment of inertia (second moment of area) measures resistance to bending about a specific in-plane axis (Ix or Iy), while the polar moment of inertia (J) measures resistance to torsion about an axis perpendicular to the cross-section. For a cross-section in the x-y plane, J = Ix + Iy. Bending analysis uses Ix or Iy; torsion analysis uses J.

What is the polar moment of inertia of a circle with diameter 5 cm?

For a solid circle with D = 5 cm (50 mm), J = π × 50⁴ / 32 = π × 6,250,000 / 32 ≈ 613,592 mm⁴ (or about 61.36 cm⁴). You can verify this directly in the calculator by setting the outer diameter to 50 mm with a solid section type.

What is the polar section modulus and how is it different from J?

The polar section modulus (Zp) is J divided by the distance c from the centroid to the outermost fiber (the outer radius). It is used to find the maximum shear stress directly: τ_max = T / Zp. While J is the geometric property, Zp conveniently combines J and c into a single design value. For a solid shaft, Zp = πD³/16.

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