Free · Physics models
Circular Section Polar Area Moment
Find polar second moment of area for a solid circle or concentric annulus from radius or diameter.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
Diameter (D) = 20.000000; Inner diameter (d) = 10.000000; Polar moment of inertia (J) = 14726.216. The worked example lists its inputs and unit conventions.
Jsolid = πR⁴/2; Jhollow = π(R⁴−r⁴)/2; D = 2R.
Worked example
Enter these known values and leave the other values blank.
- Select shape
- 1
- Radius (R)
- 10 mm
- Inner radius (Rᵢ)
- 5 mm
- Inner diameter (d)
- 10 mm
- Diameter (D)
- 20 mm
- Polar moment of inertia (J)
- 14726 mm⁴
Assumptions and limitations
- This is a geometric area moment with units length to the fourth, not a mass moment of inertia. The annulus must have positive inner radius smaller than the outer radius. Circular torsion applications require additional material and loading assumptions.
- Clear the quantity to solve an inverse; keep known quantities fixed with locks. Some inverses are singular or ambiguous and require further known values.
- Unit selectors convert the same physical quantity. Rounded reference constants and presets limit physical accuracy even when arithmetic displays more digits.
Common questions
What does this model assume?
This is a geometric area moment with units length to the fourth, not a mass moment of inertia. The annulus must have positive inner radius smaller than the outer radius. Circular torsion applications require additional material and loading assumptions.
Can I solve for an input?
Editable calculated fields support the captured inverse relationships. Clear the unknown and supply enough independent known values; an inverse must remain within the declared physical domain.
References
This is a geometric area moment with units length to the fourth, not a mass moment of inertia. The annulus must have positive inner radius smaller than the outer radius. Circular torsion applications require additional material and loading assumptions. Independently integrating r² over area with dA=2πr dr gives J=π(R⁴−Ri⁴)/2.