SpecCalcsReference calculators for the shop floor

Moment of Inertia Calculator

A solid circle with diameter 50 mm has an area moment of inertia of 306796 mm⁴, a section modulus of 12272 mm³, and a cross-sectional area of 1963 mm².

Select cross-section

Shape

Diameter (mm)

I (moment of inertia)306796 mm⁴

S (section modulus)12272 mm³

A (area)1963 mm²

The formulas

Solid circle

I = \frac{\pi d^4}{64}

Solid rectangle

I = \frac{b h^3}{12}

d diameter · b width · h height (parallel to bending axis)

The area moment of inertia (also called the second moment of area) measures how a cross-section's area is distributed relative to a bending axis. A larger I means the section resists bending more effectively — which is why I-beams concentrate material in the flanges, far from the neutral axis, rather than distributing it uniformly like a solid rectangle.

Why it matters

Beam deflection is inversely proportional to I: doubling the moment of inertia cuts deflection in half. Column buckling loads (Euler's formula) also scale with I. The section modulus S = I/c (where c is the distance to the extreme fibre) directly gives the maximum bending stress: σ = M/S. Designers compare S values to choose the lightest section that keeps stress below the allowable limit.

Hollow vs. solid sections

A hollow tube can approach the moment of inertia of a solid bar at a fraction of the weight because moving material away from the neutral axis increases I by the cube of the distance. The trade-off is wall buckling — very thin walls may buckle locally before the section reaches its theoretical bending strength.