Mass moment of inertia (kg·m²)
| Shape | Axis | I |
|---|---|---|
| Solid sphere (R) | Center | (2/5) M R² |
| Hollow sphere (R) | Center | (2/3) M R² |
| Solid cylinder (R) | Axis | ½ M R² |
| Solid cylinder (R, L) | Transverse | (1/12) M (3R² + L²) |
| Thin hoop (R) | Axis | M R² |
| Thin hoop (R) | Diameter | ½ M R² |
| Thin rod (L) | Center | (1/12) M L² |
| Thin rod (L) | End | (1/3) M L² |
| Solid rectangular plate (a × b) | Perpendicular at center | (1/12) M (a² + b²) |
| Solid cube (s) | Face-centered axis | (1/6) M s² |
Area moment of inertia (m⁴) — for beams
| Cross-section | About | I |
|---|---|---|
| Rectangle (b × h) | Neutral x | b h³ / 12 |
| Rectangle (b × h) | Neutral y | h b³ / 12 |
| Solid circle (R) | Any diameter | π R⁴ / 4 |
| Hollow circle (R_o, R_i) | Any diameter | π (R_o⁴ − R_i⁴) / 4 |
| Triangle (b, h) | Base | b h³ / 36 |
| I-beam (approx, b·t flanges, t_w·h web) | Neutral | Sum of flange + web contributions |
Parallel axis theorem
| I about offset axis | I = I_cm + M d² (mass) — I = I_c + A d² (area) |
|---|---|
| d | Distance from centroidal axis to new axis |
Notes
- Mass moment of inertia I (kg·m²) governs rotational kinetics: τ = I · α.
- Area moment of inertia I (m⁴) governs beam bending: σ = M · y / I.
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