How this calculator works
Integrating squared distance from the section centre over area gives the polar second moment. For a concentric annulus, subtract the inner circular contribution from the outer contribution. The result has length to the fourth power, not kg m2. For circular shafts it is also the Saint-Venant torsion constant; that equivalence does not generally hold for other section shapes. Use consistent radii or diameters and keep the inner size smaller than the outer size.
Formula
J=integral(r^2 dA)=pi(R^4-r^4)/2=pi(D^4-d^4)/32. 1 mm4=1e-12 m4; 1 in4=(0.0254)^4 m4.
Worked example
A solid circle of radius 1 m has J=pi/2=1.570796327 m4. An annulus with outer radius 2 m and inner radius 1 m has J=15pi/2=23.561944902 m4.
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Frequently asked questions
Is this the inertia used in rotational kinetic energy?
No. Rotational kinetic energy uses mass moment of inertia, with units kg m2. This page calculates an area property in m4.
Can I use diameter measurements?
Yes. Choose Diameters before entering outer and inner sizes.
Why does a small size error matter?
The fourth-power dependence makes the result sensitive to dimensional errors, especially for thin walls.