Area Moment of Inertia Calculator (Second Moment of Area)

The area moment of inertia (second moment of area) measures how a cross-section's material is distributed relative to the axis it bends around — the same amount of material pushed farther from that axis resists bending far more, which is why this site's Beam Deflection and Column Buckling calculators both need it, but take it as a direct input since it depends entirely on the shape and dimensions chosen. This calculator computes it for the three shapes every mechanics-of-materials reference tabulates, about each one's own centroidal bending axis: a rectangle (width b, height h) is I = b·h³/12; a circle, solid or hollow (outer diameter D, inner diameter d — enter 0 for a solid circle) is I = π·(D⁴ − d⁴)/64; a triangle (base b, height h) is I = b·h³/36.

Radius of gyration, k = √(I/A) where A is the cross-section's own area, is reported alongside I since it's the other standard figure built from the same two quantities — it's what a column's slenderness ratio (length ÷ k) is calculated from, and for a solid circle it reduces to the tidy closed form k = D/4.