Moment of Inertia Calculator

Area moment of inertia, section modulus, polar moment, and radius of gyration for 8 cross-section shapes. Includes composite section tool with parallel axis theorem.

Shape

Flange width bf (m)

Flange thickness tf (m)

Web height hw (m)

Web thickness tw (m)

Ix (about centroidal x-axis)9210.8333 cm⁴

Iy (about centroidal y-axis)845.8333 cm⁴

Sx = Ix / ȳ (elastic section modulus x)6.5792 cm²

Sy = Iy / x̄ (elastic section modulus y)1.1278 cm²

J = Ix + Iy (polar moment)10056.6667 cm⁴

rx = √(Ix/A) (radius of gyration x)11.4710 cm

ry = √(Iy/A) (radius of gyration y)3.4761 cm

A (cross-sectional area)70.0000 cm²

ȳ (centroid from baseline)14.0000 cm

What This Calculator Does

The area moment of inertia (second moment of area) Ix and Iy quantify a cross-section's resistance to bending. This calculator supports eight common structural shapes: solid rectangle, hollow rectangle/box, solid circle, hollow circle/annulus, I-beam, T-section, L-section/angle, and triangle. The composite section tool lets you combine up to five shapes (with holes/voids) and applies the parallel axis theorem automatically.

It combines Shape, Width / Flange Width (m), Height / Total Height (m), Radius (Circle only, m) to estimate Ix (centroidal x-axis), Iy (centroidal y-axis), Sx = Ix / ȳ (section modulus).

Formula & Method

Solid rectangle:

I_x = \frac{bh^3}{12}, \quad I_y = \frac{hb^3}{12}

Solid circle:

I_x = I_y = \frac{\pi r^4}{4}

I-beam (approximation):

I_x = \frac{b_f H^3 - (b_f - t_w) h_w^3}{12}

Elastic section modulus:

S_x = I_x / \bar{y}

. Parallel axis theorem:

I = I_{centroid} + A \cdot d^2

where

d

is the distance between parallel axes.

Notation used in the formulas: $R$ = Ix (centroidal x-axis); $x_{1}$ = Shape; $x_{2}$ = Width / Flange Width (m); $x_{3}$ = Height / Total Height (m); $x_{4}$ = Radius (Circle only, m).

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use this calculator for quick scenario analysis. Start with baseline values, change one driver at a time, and compare how sensitive the results are to each input shown above.

Worked Examples

Wide-flange I-beam — 150mm flange, 250mm web

bf = 150mm, tf = 15mm, hw = 250mm, tw = 10mm H = 250 + 2×15 = 280mm Ix = (150×280³ − 140×250³) / 12 = (150×21,952,000 − 140×15,625,000) / 12 = (3,292,800,000 − 2,187,500,000) / 12 = 92,108,333 mm⁴ ≈ 92.1 × 10⁶ mm⁴ Sx = 92,108,333 / 140 = 657,917 mm³ ≈ 658 cm³

Composite — rectangle with circular hole

Shape 1 (solid): 200mm × 300mm rectangle, centroid at (100, 150)mm Ix1 = 200×300³/12 = 450×10⁶ mm⁴ Shape 2 (void): circle r=50mm, offset (100, 150)mm Ix2 = π×50⁴/4 = 4.91×10⁶ mm⁴, d=0 (concentric) Composite Ix = 450×10⁶ − 4.91×10⁶ = 445.1×10⁶ mm⁴

Moment of Inertia Formulas Reference

Shape	Ix (about centroidal x-axis)	Iy
Rectangle b×h	bh³/12	hb³/12
Hollow rectangle	(bH³ − bi·hi³)/12	(hB³ − hi·bi³)/12
Circle radius r	πr⁴/4	πr⁴/4
Hollow circle	π(ro⁴ − ri⁴)/4	π(ro⁴ − ri⁴)/4
Triangle b×h	bh³/36	hb³/36

Reference Book

Engineering Mechanics: Statics & Dynamics

Russell C. Hibbeler · Pearson

A standard mechanics reference for torque, pulleys, equilibrium, load balance, and rigid-body reasoning.

View Book

Inputs Used

Shape: Used directly in the calculation.
Width / Flange Width (m): Used directly in the calculation.
Height / Total Height (m): Used directly in the calculation.
Radius (Circle only, m): Used directly in the calculation.

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