Learn: Polar second moment — solid circle
This geometric property describes how area is spread away from an axis for torsion about the centre of a solid circular section. It is not the area of the circle and it is not a mass moment of inertia.
What the formula is saying
The diameter is raised to the fourth power, making section size very influential. For this axis, integration over the solid circle gives the multiplier π/32.
Read the symbols in plain language
- D
- Diameter
Diameter. Area farther from the axis contributes strongly, producing a fourth-power size dependence.
mMetres measure length; 1 m = 1000 mm.
- J
- Result to find
Polar second moment — solid circle. Multiply by π and divide by 32 for the specified axis of the solid circle.
m⁴
Sort out the units first
Use a diameter in metres to obtain m⁴. Converting a length unit requires the fourth power: 1 m⁴ = 10¹² mm⁴. The calculator converts the diameter before applying the power.
Assumptions before calculating
The cross-section is a complete solid circle with uniform geometry. The specified axis passes through its centre; the diameter is strictly positive.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
Read the supplied values as one complete study case. Find J and explain the result in the stated output unit.
- D · Diameter
- 0.1 m
Raise the diameter to the fourth power
Area farther from the axis contributes strongly, producing a fourth-power size dependence.
(0.1)^4 = 0.0001 m⁴Apply the circular-section factor
Multiply by π and divide by 32 for the specified axis of the solid circle.
π × (0.0001) ÷ 32 ≈ 0.000009817477042 m⁴
Does this worked answer make sense?
Doubling the diameter multiplies the result by 16. For the same solid circle, the polar value is twice the value about one centroidal diameter.
A second worked example — different values
A second case uses different data. Predict which way the answer will change, then calculate it without reusing the first answer.
- D · Diameter
- 0.2 m
Raise the diameter to the fourth power
Area farther from the axis contributes strongly, producing a fourth-power size dependence.
(0.2)^4 = 0.0016 m⁴Apply the circular-section factor
Multiply by π and divide by 32 for the specified axis of the solid circle.
π × (0.0016) ÷ 32 ≈ 0.0001570796327 m⁴
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Solve this separate case yourself. Use only the values below; the two worked examples use different data. Give the requested result in the selected unit.
- D · Diameter
- 0.15 m
Find: Learn: Polar second moment — solid circle
A hint, not the answer
The diameter is raised to the fourth power, making section size very influential. For this axis, integration over the solid circle gives the multiplier π/32.
Use a diameter in metres to obtain m⁴. Converting a length unit requires the fourth power: 1 m⁴ = 10¹² mm⁴. The calculator converts the diameter before applying the power.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Raise the diameter to the fourth power
Area farther from the axis contributes strongly, producing a fourth-power size dependence.
(0.15)^4 = 0.00050625 m⁴Apply the circular-section factor
Multiply by π and divide by 32 for the specified axis of the solid circle.
π × (0.00050625) ÷ 32 ≈ 0.00004970097753 m⁴
Avoid the common trap
Use the diameter, not the radius. Do not confuse I = πD⁴/64 with J = πD⁴/32. A unit written m⁴ must not be reported as m².
When this method applies — and when it does not
For a hollow circle, subtract the inner-circle property from the outer-circle property. Do not use the solid-circle expression for a tube, or use a polar area moment as the torsion constant of a noncircular section.
For study and understanding, not approval of a real structure, site operation or design. Apply the correct standard, National Annex and professional review to actual engineering work.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Reading focus: Polar second moment — solid circle. Use the relevant elastic-response, equilibrium, constitutive-relations or shear-and-torsion module; the lesson states its particular sign convention.
Lesson updated: · Both examples and the separate practice case are checked against an independent high-precision numerical implementation. This verifies arithmetic for the stated model, not engineering certification.