Polar Moment of Inertia
Compute polar moment of inertia J = πR⁴/2 for solid circles and J = π(R⁴-Ri⁴)/2 for hollow sections. Free torsional property calculator with charts and unit conversions.
About This Calculator
About the Polar Moment of Inertia Calculator
The polar moment of inertia (J), also known as the second polar moment of area, is a fundamental geometrical property used in mechanical and civil engineering to predict the torsional resistance of circular shafts and beams. This calculator computes J for both solid and hollow circular cross-sections and presents results in four common units: mm⁴, cm⁴, m⁴, and in⁴.
For a solid circular section, the polar moment is calculated using J = πD⁴/32, where D is the outer diameter. For a hollow circular section, the formula is J = π(D⁴-d⁴)/32, where d is the inner diameter. These formulas assume the section is uniform along its length and made of a homogeneous material.
The polar moment of inertia is essential for analyzing torsionally loaded members such as drive shafts, axles, torque tubes, and steering columns. It appears directly in two key equations: the shear stress formula τ = Tρ/J (where T is torque and ρ is the radial distance from the center) and the angle of twist formula φ = TL/JG (where L is length and G is the shear modulus of the material).
Engineers use the polar moment to ensure shafts can safely transmit required torque without exceeding allowable stress limits or experiencing excessive angular deflection. A larger polar moment means greater torsional stiffness — which is why hollow shafts are often preferred over solid ones when weight reduction is important, as they provide high J with less material.
Key Formula
Solid Circle: J = (π/2) × R⁴ = (π/32) × D⁴
Hollow Circle: J = (π/2) × (R⁴ − Ri⁴) = (π/32) × (D⁴ − d⁴)
Where R = outer radius, Ri = inner radius, D = outer diameter, d = inner diameter.
Regional Notes
This calculator uses the metric system (mm) as the default input unit, which is standard worldwide for engineering calculations. The outputs include both metric (mm⁴, cm⁴, m⁴) and imperial (in⁴) units for convenience. Engineers in the US commonly use in⁴, while engineers in India, the UK, Europe, and most other regions use mm⁴. The formulas and underlying physics are universal and apply globally.
Frequently Asked Questions
What is polar moment of inertia?
Polar moment of inertia (J) is a geometrical property of a cross-section that measures its resistance to torsion (twisting). It is also called the second polar moment of area. For circular shafts, it directly relates to shear stress and angle of twist under torsional loading.
How do you calculate polar moment of inertia for a solid circle?
For a solid circular section, the polar moment of inertia is calculated using the formula J = πD⁴/32, where D is the outer diameter. In terms of radius R, it is J = πR⁴/2. The result is expressed in units of length to the fourth power, such as mm⁴, cm⁴, m⁴, or in⁴.
What is the formula for polar moment of inertia of a hollow circle?
For a hollow circular section, the polar moment of inertia is J = π(D⁴-d⁴)/32, where D is the outer diameter and d is the inner diameter. In terms of radii, it is J = π(R⁴-Ri⁴)/2. The hollow section always has a lower polar moment than a solid section of the same outer diameter.
What are the units of polar moment of inertia?
Polar moment of inertia is measured in units of length to the fourth power. Common units include mm⁴ (millimeters to the fourth), cm⁴ (centimeters to the fourth), m⁴ (meters to the fourth), and in⁴ (inches to the fourth). Our calculator shows results in all four units simultaneously.
What is the difference between polar moment of inertia and area moment of inertia?
The polar moment of inertia (J) measures resistance to torsion (twisting) about the longitudinal axis, while the area moment of inertia (I) measures resistance to bending. For circular sections, J equals the sum of the two perpendicular area moments (J = Ix + Iy). Polar moment is used in torsion calculations, while area moment is used in bending calculations.
Why is polar moment of inertia important in engineering?
Polar moment of inertia is critical in mechanical and civil engineering for designing shafts, axles, and other torsionally loaded circular members. It appears in the shear stress formula τ = Tρ/J and the angle of twist formula φ = TL/JG. A higher polar moment means lower shear stress and less angular deflection under a given torque.
Is the polar moment of inertia calculator free?
Yes, this polar moment of inertia calculator is completely free to use with no registration required. It works for both solid and hollow circular sections and instantly computes results in mm⁴, cm⁴, m⁴, and in⁴ with interactive charts.