Moment of inertia calculator
Eleven parametric cross-sections: rectangle, hollow box, solid and hollow circle, semicircle, I-beam, T-beam, channel, L-angle, triangle, trapezoid. Returns Ix, Iy, polar J, section moduli S and Z, radii of gyration, area, centroid. Parallel-axis theorem in one click. Live cross-section drawing with centroidal axes marked. Reviewed by a licensed PE.
Use the calculator
Pick a cross-section, enter the dimensions, switch units, optionally compute about an offset (parallel) axis. The drawing shows your section with the centroidal x-y axes drawn in.
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 moment of inertia formula
Moment of inertia (also called the second moment of area in mechanics) of a cross-section A about a chosen axis is the integral of squared distance from that axis, weighted by area. For an axis through the centroid, this gives the smallest possible value — the "centroidal" moment of inertia, the one that appears in the bending stiffness expression EI.
- I_x
- second moment of area about x-axis, mm⁴ / in⁴
- I_y
- second moment of area about y-axis, mm⁴ / in⁴
- J
- polar moment of inertia, mm⁴ / in⁴
- A
- cross-sectional area, mm² / in²
- y
- distance from the x-axis, mm / in
- r
- distance from the polar axis, mm / in
For standard shapes, the integral has been evaluated once and the result reduced to a closed-form expression in the dimensions of the section. The calculator above and the Variants section below tabulate those closed forms for the eleven most common shapes.
- I
- I about the offset (parallel) axis, mm⁴
- I_c
- I about the centroidal axis, mm⁴
- A
- area, mm²
- d
- perpendicular distance between the two axes, mm
The parallel-axis theorem is what lets composite sections (T-beams, L-angles, channels, custom built-up sections) be calculated by adding contributions of each simple sub-shape, each shifted from the overall centroid by its own d.
Worked example, 100 × 200 rectangle
A solid rectangular cross-section, 100 mm wide × 200 mm tall. Compute every common property by hand so you can verify the calculator.
| Quantity | Formula | Substitution | Result |
|---|---|---|---|
| Area A | b · h | 100 × 200 | 20 000 mm² |
| Ix | b·h³/12 | 100 × 200³ / 12 | 6.67 × 10⁷ mm⁴ |
| Iy | h·b³/12 | 200 × 100³ / 12 | 1.67 × 10⁷ mm⁴ |
| Polar J | Ix + Iy | 6.67e7 + 1.67e7 | 8.33 × 10⁷ mm⁴ |
| Sx | Ix / (h/2) | 6.67e7 / 100 | 6.67 × 10⁵ mm³ |
| rx | √(Ix/A) | √(6.67e7 / 20 000) | 57.7 mm |
| I about bottom edge | Ix + A·d² with d = 100 | 6.67e7 + 20 000 · 100² | 2.67 × 10⁸ mm⁴ |
The "I about bottom edge" line equals b·h³/3 by direct integration — a quick sanity check that the parallel-axis theorem and the centroidal formula are consistent.
Variants and special cases (formula reference)
Closed-form moments of inertia for the eleven cross-sections this calculator supports, plus a few related concepts the same vocabulary covers.
Rectangle
Ix = b·h³/12 about centroidal axis parallel to base; Iy = h·b³/12; rx = h/√12. Square is a special case b = h.
Hollow rectangle / box section
For outer b × h and centred inner cavity bᵢ × hᵢ: Ix = (b·h³ − bᵢ·hᵢ³)/12. The same approach extends to wall thickness inputs by setting bᵢ = b − 2t, hᵢ = h − 2t.
Solid circle and pipe
Solid: Ix = Iy = π·D⁴/64; J = π·D⁴/32. Hollow (pipe): Ix = π(D⁴ − d⁴)/64. Round shafts in pure torsion use J directly in the formula τ = T·r/J.
Semicircle
About horizontal centroidal axis: Ix = (π/8 − 8/(9π))·r⁴ ≈ 0.1098·r⁴. Centroid sits at ȳ = 4r/(3π) above the flat edge — almost half the radius.
I-beam (W / IPE / UB)
Treat the section as the outer bounding rectangle minus two rectangular notches: Ix = [b·h³ − (b − tw)·(h − 2tf)³]/12. AISC W-shapes and European IPE/UB use this form; the dedicated I-beam mode in the calculator handles flange and web thicknesses directly.
T-beam
Composite of a top flange (b × tf) and a vertical stem (tw × hw). Centroid is below the flange; calculate the centroid first, then sum each rectangle's Icentroidal + A·d². The T-beam mode of the calculator uses this composite procedure internally.
Channel (C-section)
Open section with two flanges on the same side of the web. Asymmetric, so the centroid is offset from the web. Compose three rectangles and sum with parallel-axis offsets — the calculator's c_channel mode does this directly.
L-angle
Two perpendicular legs meeting at a corner, uniform thickness. Centroid is offset from both edges. Same composite-with-parallel-axis procedure; for equal legs (a = b), the centroid sits along the symmetry diagonal at (x̄, x̄).
Right triangle
Base b, height h, right angle at corner. Centroid at (b/3, h/3). Ix = b·h³/36 about centroidal axis parallel to base; about an axis at the base, I = b·h³/12 by parallel-axis theorem.
Trapezoid (isosceles)
T