Section PropertiesIx · Iy · S · rMetric / ImperialFree

Moment of Inertia
Calculator

The second moment of area (I) drives bending stiffness and deflection. Pick a cross-section, enter its dimensions, and get Ix, Iy, area, section modulus (S = I/c), and radius of gyration (r = √(I/A)) — the inputs to deflection (δ ∝ 1/I) and bending-stress (Fb = M/S) checks.

Who it's for: Structural engineers and designers entering bending-stiffness inputs into deflection and stress checks — bring the cross-section dimensions and choose mm, cm, or in.

Choose the cross-section

Rectangle, solid circle, circular tube, or symmetric I-beam / wide-flange

Enter dimensions + units

All dimensions in the same unit (mm, cm, or in)

Get I, S, r

Strong-axis Ix, weak-axis Iy, area A, section modulus Sx, radius of gyration rx

I = second moment of area

Units are length⁴ (mm⁴, in⁴). Section modulus S = I / c (c = distance to extreme fiber); radius of gyration r = √(I / A). These feed deflection (δ = 5wL⁴/384EI etc.) and bending stress (fb = M/S).

Strong-axis formulas

Section	Moment of inertia
Rectangle (b×h)	Ix = b·h³ / 12
Solid circle (D)	I = π·D⁴ / 64
Circular tube (D, d)	I = π·(D⁴ − d⁴) / 64
I-beam (H, B, tf, tw)	Ix = [B·H³ − (B−tw)(H−2tf)³] / 12

Section Calculator

Ix · Iy · A · Sx · rx

How to use this calculator

Choose the cross-section shape

Select Rectangle, solid circle, circular tube, or I-beam / wide-flange. The dimension fields update automatically when you change shape.

Select units

Pick mm, cm, or in. Every dimension field uses the same unit — mixing mm and m inflates the result by many orders of magnitude.

Enter the width or outer diameter

For a rectangle enter b (base width). For circles and tubes enter outer diameter D; for a tube, also enter inner diameter d.

Enter depth and remaining dimensions

For a rectangle enter h (the bending depth, the tall dimension). For an I-beam enter H (overall depth), B (flange width), tf (flange thickness), and tw (web thickness).

Read Ix, Sx, and rx

The result panel shows strong-axis Ix and weak-axis Iy in length⁴ units, cross-sectional area A, section modulus Sx = Ix / c, and radius of gyration rx = √(Ix / A).

The formula

The moment of inertia I measures how far a cross-section's area is spread from its neutral axis — area farther from the axis contributes more to bending stiffness, so a taller section is stiffer than a wider one.

Rectangle (strong axis): Ix = b · h³ / 12
Solid circle: I = π · D⁴ / 64
Circular tube: I = π · (D⁴ − d⁴) / 64
I-beam / wide-flange: Ix = [B · H³ − (B − tw) · (H − 2tf)³] / 12

b = base width; h = depth in bending; D = outer diameter; d = inner diameter;
H = overall I-beam depth; B = flange width; tf = flange thickness; tw = web thickness

Derived properties (all shapes):
Sx = Ix / c — section modulus; c = distance to extreme fiber (h/2 for symmetric sections)
rx = √(Ix / A) — radius of gyration; A = cross-sectional area

Worked example

Example

A timber joist is 200 mm wide and 400 mm deep (rectangle, units = mm). The strong-axis moment of inertia is Ix = 200 × 400³ ÷ 12 = 1.067 × 10⁹ mm⁴. With c = 200 mm, section modulus Sx = 1.067 × 10⁹ ÷ 200 = 5,333,333 mm³. Area A = 80,000 mm², so radius of gyration rx = √(1.067 × 10⁹ ÷ 80,000) = 115 mm.

When this estimate will be off

Assumes a uniform cross-section along the full length. Tapered members and sections with notches need a separate calculation at each critical point.
Computes the gross elastic moment of inertia only. For cracked reinforced-concrete members, use the effective Ie per ACI 318 §24.2 — the gross I overestimates stiffness after cracking.
Section modulus Sx uses c = half the depth, correct only for symmetric sections. For unsymmetric cross-sections, c must be the actual distance to the farther extreme fiber and two S values are needed.
I and S are geometric inputs to design checks, not design capacities. Deflection, bending-stress, and buckling checks require applying these values to code equations with material properties and load factors.

Frequently asked questions

What is the moment of inertia used for in structural design?

It controls bending stiffness and deflection. For a simply supported beam with uniform load, mid-span deflection δ = 5wL⁴ / (384EI) — doubling I halves the deflection for the same load, span, and material. It also feeds the bending-stress check: the elastic stress at the extreme fiber is f_b = M / S_x, where S_x = I_x / c.

What is the difference between elastic and plastic section modulus?

This calculator gives the elastic section modulus S_x = I_x / c. The plastic section modulus Z_x, used in AISC Chapter F plastic-moment checks, is different — for a rectangle Z_x = b · h² / 4, which is 1.5 × S_x. AISC shape tables list both; for a custom section, calculate Z_x separately from the plastic neutral axis.

What is the radius of gyration and when do I need it?

r_x = √(I_x / A). It appears in column buckling checks — the slenderness ratio KL / r compares the effective unbraced length to how spread the cross-section area is. A higher r means a more buckling-resistant column. For beams, the weak-axis r_y governs lateral-torsional buckling.

Why does the I-beam formula subtract from a solid rectangle?

It uses the standard hollow-subtraction method: compute I for a solid B × H rectangle, then subtract I for the interior void — a rectangle of width (B − tw) and height (H − 2tf) representing the open space between the flanges outside the web. The result equals the integral ∫y² dA across the actual I cross-section and matches published AISC table values.

Sources

AISC Steel Construction Manual, Part 1 — tabulated Ix, Iy, Sx, Sy, rx, ry for all standard W, S, C, and HSS sections
ACI 318-19 §24.2 — effective moment of inertia for deflection computation of cracked reinforced-concrete members

This free Calculator is built and maintained by DataDrivenAEC, using the relevant codes and standards. It does not substitute for professional judgment.