inch to the fourth
Area moment of inertiaexact by definition
The inch to the fourth power is exactly 4.162314256 × 10⁻⁷ m⁴, since the inch is exactly 0.0254 m and that fourth power terminates exactly. It is the standard unit in US steel and timber design, where AISC lists \(I_x\) and \(I_y\) in in⁴: a W12×26 has \(I_x\) of about 204 in⁴.
| 1 in⁴ | 416,231.43 mm⁴ |
About the inch to the fourth
In US design the second moment of area in in⁴ is what sits in the denominator of every deflection formula. A simply supported beam under a uniform load deflects \(\delta = 5wL^4/(384EI)\), and with \(E\) for steel at 29 000 ksi, lengths in inches and \(w\) in kip per inch, the answer comes out in inches with no conversion needed. That internal consistency is why the customary system's structural corner works despite the mess elsewhere: kips, inches, ksi and in⁴ are a coherent set.
The fourth-power dependence on depth is where the design intuition lives. A 2×10 joist laid on edge is roughly 3.4 times stiffer than a 2×8 in the same span, from depth alone, so a floor that bounces is almost always solved by going deeper rather than wider. The same logic drives the shape of every rolled section: material near the neutral axis contributes almost nothing to \(I\), so an efficient bending member is thin in the middle and heavy at the extremes, constrained only by the need to keep the thin web from buckling.