Area Moment of Inertia — Rectangle

The second moment of area measures how far a section's material sits from the axis it bends about — and because the contribution goes as distance squared, it rewards depth enormously. For a rectangle, I = bh³/12 about the axis through the centroid perpendicular to h. A 50 mm × 100 mm bar bending about its strong axis has I = 0.05 × 0.1³ ÷ 12 = 4.167 × 10⁻⁶ m⁴; turn it flat and h and b swap, giving 1.042 × 10⁻⁶ m⁴, a quarter as stiff. Double the depth alone and stiffness goes up eightfold — the whole reason joists stand on edge and I-beams put their steel in the flanges.

This engine has no m⁴ unit type, so I is entered and returned as a plain number in m⁴: 1 in⁴ = 4.162314 × 10⁻⁷ m⁴ and 1 mm⁴ = 10⁻¹² m⁴. The trap is the cube: get b and h the wrong way round and you are not off by a little, you are off by (h/b)⁴ — for a 2 × 8 that is a factor of 16. And this formula is only valid about the centroidal axis; if the rectangle is a flange offset from the neutral axis of a built-up section, you must add the parallel-axis term Ad².