Method
For a rectangle, Ix = bh3/12 and Iy = hb3/12. For a solid circle, I = πd4/64 and J = πd4/32.
Worked example
A 50 mm × 100 mm rectangle has area 5,000 mm² and strong-axis second moment of area 4.167 × 106 mm4.
Why orientation matters
The same 50 × 100 mm timber is much stiffer on edge than flat. Bending stiffness is proportional to I, and I grows with the cube of the depth in the bending direction:
| Section (b × h, mm) | Area (mm²) | Ix (×106 mm4) | Section modulus Zx (×103 mm³) |
|---|---|---|---|
| 100 × 50 | 5,000 | 1.042 | 41.7 |
| 50 × 100 | 5,000 | 4.167 | 83.3 |
| 50 × 150 | 7,500 | 14.063 | 187.5 |
| 50 × 200 | 10,000 | 33.333 | 333.3 |
| 100 × 100 | 10,000 | 8.333 | 166.7 |
Turning the 50 × 100 mm section on edge (100 mm deep) instead of flat (50 mm deep) multiplies I by (100 / 50)² = 4 for the same area, so the beam deflects a quarter as much under the same load. Bending stress is M / Z, so the section modulus Z = I / (h / 2) sets the stress at the outer fibre.
Units
Second moment of area has units of length to the fourth power. 1 cm4 = 104 mm4 and 1 m4 = 1012 mm4; steel section tables often list values in cm4, so check the unit before comparing with a result in mm4.
Important limits
These are centroidal properties for ideal solid shapes only. Do not use them directly for a hollow, built-up, composite, offset or perforated section without calculating the correct geometry.
Related tools
Euler Buckling Calculator · Cantilever Beam Deflection Calculator · Mechanics & Materials Calculators