Area moment of inertia: section formulas and calculator
Published October 12, 2026 — the area moment of inertia and section modulus of common sections (rectangle, round, tubes, I-section, triangle), worked examples, the parallel axis theorem, unit conversions and a calculator.
In short: the area moment of inertia I (in mm⁴) measures a section's bending stiffness; it depends only on its shape. Rectangle: I = b·h³/12, with h in the bending direction. Solid round: I = π·D⁴/64. The section modulus W = I / c (in mm³) gives the stress: σ = M / W. Doubling the depth of a rectangular beam makes it 8 times stiffer.
1. Area moment of inertia calculator
Pick the section and enter its dimensions in millimetres. x axis: horizontal centroidal axis, bending acting in the direction of H (or h, D).
2. Formulas for common sections
Moments of inertia about the centroidal x axis (through the centroid), bending acting in the direction of h, H or D. W = I / c, c being the distance from the axis to the extreme fibre. J: polar moment of inertia, for the torsion of round sections.
3. Worked examples
Dimensions in millimetres. The same 20 × 40 rectangle is four times stiffer on edge than flat: the moment of inertia varies with the cube of the depth.
Rolled profiles (IPE, HEA…) have root fillets between web and flanges, which the formula ignores: their actual moment of inertia is slightly higher. Use the manufacturer's catalogue value.
4. Using I and W: stress, deflection, parallel axis
- Bending stress: σ = M / W, with M the bending moment (N·mm) and W in mm³; σ in MPa. Compare it with the material's yield strength, see S235, S275, S355 steels.
- Deflection: it is inversely proportional to E·I. Cantilever with an end load: f = F·L³ / (3·E·I); simply supported beam with a mid-span load: f = F·L³ / (48·E·I). E is 210,000 MPa for steel, about 70,000 MPa for aluminium.
- Parallel axis theorem: about an axis parallel to the centroidal axis, at distance d, I = Ic + A·d². This is how a built-up section (several rectangles) is calculated: add the I of each part, moved to the common axis.
- Orientation: depth counts cubed. For the same mass, a tube or an I-section is much stiffer than a solid round, because the material is far from the axis.
- In SOLIDWORKS, Tools → Evaluate → Section Properties gives the area, centroid and moments of inertia of a planar face or sketch: handy for any section.
5. Units and conversions
The moment of inertia is expressed in mm⁴ or cm⁴ (profile catalogues), the section modulus in mm³ or cm³. Exact conversions, 1 inch being 25.4 mm:
Printable version: the area moment of inertia formulas as a PDF (A4, no sign-up), to keep with your notes or in the design office.
6. FAQ
What is the difference between area moment of inertia and mass moment of inertia?
The area moment of inertia (second moment of area) depends only on the shape of the cross-section and is expressed in mm⁴: it is used for bending and deflection. The mass moment of inertia is expressed in kg·m² and describes a body's resistance to angular acceleration.
What is the moment of inertia formula for a rectangle?
I = b·h³/12 about the centroidal axis parallel to b, where h is the dimension in the bending direction. A 20 × 40 bar on edge (h = 40) has I = 106,667 mm⁴, four times more than flat (26,667 mm⁴).
What is the formula for a round bar and a tube?
Solid round: I = π·D⁴/64 and section modulus W = π·D³/32. Tube: I = π·(D⁴ − d⁴)/64. The polar moment, used in torsion, is twice as much: J = π·D⁴/32 for a solid round.
What is the section modulus W?
It is the moment of inertia divided by the distance from the neutral axis to the extreme fibre: W = I / c, in mm³. The maximum bending stress is σ = M / W, where M is the bending moment.
How do you calculate the moment of inertia about another axis?
With the parallel axis theorem: I = Ic + A·d², where Ic is the moment of inertia about the centroidal axis, A the section area and d the distance between the two axes.