The area moment of inertia, or second moment of area, measures how a cross-section's area is distributed relative to an axis.

For bending about the $x$ axis,

$$I_x=\int_A y^2,dA.$$

Area farther from the axis contributes more strongly because its distance is squared.

Geometry, not mass

This quantity depends only on cross-sectional geometry. It is different from the mass moment of inertia used in rotational dynamics.

Simple sections

For a rectangle of width $b$ and height $h$ about its centroidal horizontal axis,

$$I=\frac{bh^3}{12}.$$

The cubic dependence on height explains why deep beam sections resist bending so effectively.

Parallel-axis theorem

If a centroidal axis is shifted by distance $d$ to a parallel axis,

$$I=I_c+Ad^2.$$

This makes composite cross-sections straightforward to analyse by adding or subtracting simpler shapes.