Area moment of inertia of a circle
The formula π·D⁴/64 for solid circles and π(D⁴ − d⁴)/64 for hollow. Polar moment for shaft torsion. Worked examples, AISC HSS round sections, and a calculator that handles both forms. Reviewed by a licensed PE.
Use the circle moment of inertia calculator
Calculator opens with the solid circle form selected. To compute a pipe (hollow circle), switch the shape dropdown to "Hollow circle / pipe". For AISC HSS round sections, pick from the "Standard section" dropdown — dimensions auto-fill.
Pick an AISC W / IPE / HEA / HEB / HSS / channel / angle to auto-fill dimensions, or leave on "Manual entry" for a custom section.
Rectangle: simplest case. Ix = b·h³/12 about centroidal axis parallel to base.
Hollow rectangle (centred). Ix = (b·h³ − bi·hi³)/12. Ensure inner dims < outer.
Solid circle. Ix = Iy = π·D⁴/64. Polar J = π·D⁴/32.
Hollow circle / pipe. Ix = π(D⁴ − d⁴)/64. d = inside diameter.
Half-disc with flat edge at bottom. Centroid sits 4r/(3π) above the flat edge.
I-beam (W / IPE / UB). Symmetric flanges and centred web.
T-beam: top flange + vertical stem. Centroid is below the flange — origin is bottom of stem.
C-channel: web on left, flanges open to the right. Asymmetric — centroid offset from web.
L-angle: legs of width leg1 (horizontal) and leg2 (vertical), uniform thickness t.
Right triangle: right-angle at the origin, base b along x, height h along y. Centroid at (b/3, h/3).
Isosceles trapezoid: a = top, b = bottom, h = height. ȳ = h(b + 2a)/(3(a+b)) from bottom.
- Polar J
- — mm⁴
- Sx (elastic)
- — mm³
- Sy (elastic)
- — mm³
- Zx (plastic)
- — mm³
- rx
- — mm
- ry
- — mm
- Area
- — mm²
- Centroid (x̄, ȳ)
- —
- I about offset
- —
The circle moment of inertia formula
Two closed forms cover every circular cross-section. The first is for solid bars and discs; the second extends it to hollow pipes by subtracting the inner-diameter contribution.
- I_x
- second moment of area about horizontal centroidal axis, mm⁴ / in⁴
- I_y
- about vertical centroidal axis (equal to Ix), mm⁴ / in⁴
- J
- polar moment of inertia (used in torsion), mm⁴ / in⁴
- D
- outer diameter, mm / in
- D
- outer diameter, mm / in
- d
- inner diameter (set d = 0 to recover the solid case), mm / in
Both formulas are derived from the polar-coordinate integral Ipolar = ∫∫ r²·r dr dθ over the cross-section. Rotational symmetry means Ix = Iy, so polar splits into equal halves.
Worked example, 100 mm solid bar vs 100 × 80 mm pipe
Compare a solid 100 mm round bar against a pipe of the same outer diameter with an 80 mm inner diameter (i.e. wall thickness 10 mm). Same outer envelope, but the pipe removes the central 64% of area — yet retains most of the bending stiffness.
| Property | Solid 100 mm | Pipe 100 × 80 mm | Pipe / Solid |
|---|---|---|---|
| Area A | 7854 mm² | 2827 mm² | 36% |
| Ix = π·(D⁴ − d⁴)/64 | 4.91 × 10⁶ mm⁴ | 2.91 × 10⁶ mm⁴ | 59% |
| Sx = Ix / (D/2) | 9.82 × 10⁴ mm³ | 5.8 |