Second Moment of Area (Area Moment of Inertia) Formula Sheet
Reference formulas for the second moment of area of common cross-sectional shapes in engineering and structural analysis
Second Moment of Area Formulas
| Shape | Formula (about centroidal axis) |
|---|---|
| Rectangle (base b, height h) | Ix = (b·h³) / 12 |
| Circle (radius r) | I = (π·r⁴) / 4 |
| Hollow Circle (outer radius R, inner radius r) | I = (π/4)·(R⁴ − r⁴) |
| Triangle (base b, height h) | Ix = (b·h³) / 36 |
| Thin-Walled Tube (radius r, thickness t) | I ≈ π·r³·t |
| I-Beam (flange width b, height h, web width bi, web height hi) | I ≈ (b·h³ / 12) − (bi·hi³ / 12) |
| T-Beam (flange width bf, flange thickness tf, web width bw, web height hw) | I ≈ (bf·tf³ / 12) + (bw·hw³ / 12) |
| C-Channel (flange width b, height h, web thickness t) | I ≈ (b·h³ / 12) − ((b−t)·(h−t)³ / 12) |
| L-Angle (leg lengths a, b; thickness t) | I ≈ (a·t³ / 3) + (b·t³ / 3) |
| Hollow Rectangle / Box Section (outer width b, outer height h, wall thickness t) | I = (b·h³ / 12) − ((b − 2·t)·(h − 2·t)³ / 12) |
| Solid Square (side a) | I = a⁴ / 12 |
| Solid Hexagon (side a) | I = (5·√3 / 16)·a⁴ |
About the Second Moment of Area (Area Moment of Inertia) Formula Sheet
This page provides a complete reference of second moment of area (area moment of inertia) formulas for a wide range of structural and mechanical shapes. Engineers and designers can quickly determine I for beams, shafts, columns, and load-bearing components, which is essential for predicting bending, deflection, and stress distribution.
Shapes covered include rectangles, circles, triangles, hollow sections, I-beams, T-beams, C-channels, L-angles, box sections, solid squares, and hexagons. These formulas are used in structural mechanics, mechanical design, civil engineering, and materials analysis.
Applications of This Reference
- Quickly find centroidal second moments of area (I) for common geometric shapes.
- Supports accurate beam bending, shaft deflection, and column buckling calculations.
- Useful for mechanical design, structural engineering, and CAD modeling.
- Includes both solid and hollow sections, with approximate formulas for complex profiles like I-beams and T-beams.
- Enables faster load analysis and structural calculations for engineers and students alike.
By using this second moment of area formula sheet, professionals can ensure structural safety, optimize material usage, and streamline engineering analysis in mechanical, civil, and structural projects. This page serves as a reliable engineering reference for area moment of inertia calculations.