If you're searching for how to calculate the polar moment of inertia (also known as the second polar moment of area) of a circular beam subjected to torsion, you're in the right place. This polar moment of inertia calculator finds the polar moment J of a solid circle or a hollow cylinder in both the metric (SI) and the American (US customary) systems of units.
The polar moment of inertia and the second moment of area are two of the most critical geometrical properties in beam analysis. The polar moment is essential for analyzing circular elements subjected to torsion (also known as shafts), while the area moment of inertia is for parts subjected to bending.
Why do we need the polar moment of inertia?
Torsion-subjected members are widely present in engineering applications involving power transmission. The most common is the driveshaft in automobile drivetrains used to transmit power to the drive wheels. Similarly, transmission shafts are used in power generation to send the energy from turbines to electric generators. Independently of the amount of transmitted power, it'll be mandatory to calculate the stresses and deformations in those shafts to avoid mechanical failure — and for that, you'll need the polar moment.
The polar moment of inertia relates to the stress and resistance to deformation in beams subjected to a torsional load. In circular beams, we can connect it mathematically to the shear stress and the angle of twist in the following ways:
- Shear stress due to torsion in a circular shaft:
τ = T·ρ / J
where J is the polar moment of inertia, T is the torque applied to the beam, ρ is the radial distance from the shaft axis to the point of interest, and τ is the shear stress at that radial distance. - Angle of twist due to torsion in a circular shaft:
φ = T·L / (J·G)
where φ is the angle of twist (in radians), L is the shaft length, and G is the shear modulus of the shaft material.
🙋 Once you've calculated the polar moment of inertia, you can use it in our shear stress calculator to find the stress caused by a torque applied to a circular beam, or in our angle of twist calculator to obtain φ. If you're dealing with bending instead, the section modulus calculator can be helpful.
Formula: solid circle polar moment of inertia
For a solid circle (a solid circular shaft) of diameter D, the polar moment of inertia equals:
J = (π / 32) · D⁴
Equivalently, in terms of the radius R = D/2, this is J = (π / 2) · R⁴. The polar moment grows with the fourth power of the diameter, so even a small increase in diameter dramatically increases a shaft's resistance to torsion.
Formula: hollow cylinder polar moment of inertia
For a hollow cylinder (a hollow circular shaft) with an outer diameter D and an inner diameter d, the polar moment of inertia equals:
J = (π / 32) · (D⁴ − d⁴)
Because the material near the axis contributes very little to the polar moment (its radial distance is small), hollow shafts offer almost the same torsional resistance as solid ones while using far less material — which is why driveshafts are often tubular.
Why can't we use these formulas for noncircular beams?
The polar moment of inertia in the form J = ∫ ρ² dA correctly predicts torsional behavior only for circular cross-sections, where plane sections remain plane during twisting. For noncircular sections (squares, rectangles, I-beams, etc.), the cross-section warps out of plane under torsion, and you must use the torsional constant instead of the polar moment of inertia to relate torque to the angle of twist.
How do I calculate the polar moment of inertia?
To calculate the polar moment of inertia with this tool:
- Choose your unit system — metric (mm) or American/Imperial (in).
- Select the cross-section shape: solid circle or hollow cylinder.
- Enter the outer diameter D (and the inner diameter d for a hollow cylinder).
- Read off the polar moment of inertia J, plus the polar section modulus, cross-sectional area, and polar radius of gyration.
By hand, simply substitute the diameter(s) into J = (π/32)·D⁴ for a solid circle or J = (π/32)·(D⁴ − d⁴) for a hollow cylinder.
Polar moment of inertia units
| Quantity | Metric (SI) | American (USCS) |
|---|---|---|
| Diameter (D, d) | millimeter (mm) | inch (in) |
| Polar moment of inertia (J) | mm⁴ (also m⁴) | in⁴ |
| Polar section modulus (Zp) | mm³ | in³ |
| Cross-sectional area (A) | mm² | in² |
| Polar radius of gyration (k) | mm | in |
FAQs
What is the polar moment of inertia?
The polar moment of inertia (or second polar moment of area), J, is a geometric property of a cross-section that measures its resistance to torsion (twisting). For a circular cross-section it is defined as J = ∫ ρ² dA, where ρ is the radial distance from the centroidal axis.
What is the polar moment of inertia of a solid circle?
For a solid circle of diameter D, the polar moment of inertia is J = π·D⁴/32, which is equal to π·R⁴/2 in terms of the radius R = D/2.
What is the polar moment of inertia of a hollow cylinder?
For a hollow circular shaft with outer diameter D and inner diameter d, the polar moment of inertia is J = π·(D⁴ − d⁴)/32.
What is the difference between the polar moment of inertia and the area moment of inertia?
The polar moment of inertia (J) describes resistance to torsion about the longitudinal axis and is used for shafts. The area moment of inertia (I) describes resistance to bending about a transverse axis. For a circular section they are related by J = 2I, since Ix = Iy and J = Ix + Iy.
Can I use the polar moment of inertia for a square or rectangular shaft?
No. The simple polar-moment formulas are valid only for circular sections. Noncircular sections warp under torsion, so you must use the torsional constant instead to compute the angle of twist or shear stress.