Moment of Inertia Calculator

Moment of Inertia Calculator

The second moment of area measures how a cross-section's material is distributed relative to a bending axis - a critical input for beam deflection and stress calculations.

Formulas

Rectangle: I = (Width × Height³) / 12
Circle: I = (π × Radius⁴) / 4

Example

A 50mm x 100mm rectangular cross-section:

I = (50 x 100³) / 12 = 4,166,666.67 mm⁴

Why Shape Orientation Matters So Much

The same cross-sectional area arranged differently can have dramatically different resistance to bending - a tall thin rectangle standing on edge has a much higher moment of inertia (and resists bending far better) than the same material laid flat as a wide, short rectangle, even though both use identical material area.

The Logic Behind I-Beam Design

This is exactly why structural I-beams concentrate material at the top and bottom flanges, far from the neutral bending axis at the center - this arrangement maximizes moment of inertia (and therefore bending resistance) for a given total amount of material used, which is far more material-efficient than a solid rectangular cross-section of equivalent bending strength.