Civil & Structural Engineering

Area Moment of Inertia Calculator

Calculate centroidal Ix, Iy, area, section modulus, and radius of gyration for common beam cross-sections.

Formula Ix = b h³ / 12Reviewed Aug 27, 2026

Area moment of inertia, also called the second moment of area, measures how a cross-section's area is distributed around a bending axis. It is the I used in beam deflection, bending stress, and column buckling formulas, with units such as mm⁴ or in⁴. This calculator covers common idealized sections and reports both strong-axis and weak-axis values so users do not confuse Ix with Iy or with mass moment of inertia.

Calculation Bench

Cross-Section

01Width

b · Overall width of a rectangular section.

02Height / Overall Depth

h · Overall depth used as the strong-axis bending dimension.

Solution

Choose a section shape, enter its dimensions, and calculate centroidal area moments of inertia.

Ix = b h³ / 12; Iy = h b³ / 12

Formula Sheet

I_x = \dfrac{b h^3}{12},\quad I_y = \dfrac{h b^3}{12}

I_x = \dfrac{B H^3 - b h^3}{12},\quad I_y = \dfrac{H B^3 - h b^3}{12}

I_x = I_y = \dfrac{\pi D^4}{64}

I_x = I_y = \dfrac{\pi(D^4 - d^4)}{64}

I_x = \dfrac{B H^3 - (B - t_w)(H - 2t_f)^3}{12},\quad I_y = \dfrac{2t_f B^3 + (H - 2t_f)t_w^3}{12}

S = \dfrac{I}{c},\quad r = \sqrt{\dfrac{I}{A}}

IxArea Moment of Inertia about x-axis
IyArea Moment of Inertia about y-axis
ACross-Sectional Area
SxSection Modulus about x-axis
SySection Modulus about y-axis
rxRadius of Gyration about x-axis
ryRadius of Gyration about y-axis
bWidth
hHeight / Depth
tWall Thickness
DOuter Diameter
t_fFlange Thickness
t_wWeb Thickness

Variables & Units

Symbol	Variable	Description	Common Units
Ix	Area Moment of Inertia about x-axis	Second moment of area about the horizontal centroidal axis.	mm⁴, cm⁴, m⁴, in⁴, ft⁴
Iy	Area Moment of Inertia about y-axis	Second moment of area about the vertical centroidal axis.	mm⁴, cm⁴, m⁴, in⁴, ft⁴
A	Cross-Sectional Area	Geometric area of the selected cross-section after subtracting voids.	mm², cm², m², in², ft²
Sx	Section Modulus about x-axis	Elastic section modulus derived from Ix and the vertical extreme-fiber distance.	mm³, cm³, m³, in³
Sy	Section Modulus about y-axis	Elastic section modulus derived from Iy and the horizontal extreme-fiber distance.	mm³, cm³, m³, in³
rx	Radius of Gyration about x-axis	rx = √(Ix/A), useful in buckling and slenderness checks.	mm, cm, m, in, ft
ry	Radius of Gyration about y-axis	ry = √(Iy/A), useful in weak-axis buckling checks.	mm, cm, m, in, ft
b	Width	Overall width of a rectangular section.	mm, cm, m, in, ft
h	Height / Depth	Overall depth used as the strong-axis bending dimension.	mm, cm, m, in, ft
t	Wall Thickness	Uniform thickness for hollow rectangular or circular sections.	mm, cm, m, in, ft
D	Outer Diameter	Outside diameter of a solid or hollow circular section.	mm, cm, m, in, ft
t_f	Flange Thickness	Thickness of each flange in the symmetric I-section model.	mm, cm, m, in, ft
t_w	Web Thickness	Thickness of the web in the symmetric I-section model.	mm, cm, m, in, ft

How to Use This Calculator

01Choose the cross-section shape: rectangle, hollow rectangle, solid circle, hollow circle/tube, or symmetric I-section.
02Enter the section dimensions. Hollow sections use outer size plus wall thickness; the I-section uses overall depth, flange width, flange thickness, and web thickness.
03Select Calculate to get Ix and Iy about the centroidal axes, plus area, section modulus, and radius of gyration.
04Use Ix for strong-axis beam deflection or bending checks when the vertical depth is the bending dimension; use Iy for weak-axis checks.
05Compare shapes carefully: these formulas are idealized geometric section properties, not full structural steel table values with fillets and manufacturing tolerances.

How the Formula Works

The second moment of area is a geometric integral of distance squared over area. Material placed far from the neutral axis contributes much more than material near the center, which is why increasing section depth can increase stiffness dramatically. For a rectangle, Ix = b h³ / 12, so depth is cubed for bending about the horizontal centroidal axis.

Hollow sections are calculated by subtracting the inner void from the outer section. A circular tube uses Ix = Iy = π(D⁴ − d⁴)/64, while a rectangular tube uses separate formulas for Ix and Iy because width and height are usually different.

The calculator also reports elastic section modulus S = I/c and radius of gyration r = √(I/A). These secondary values help connect the same geometry to bending stress and column buckling workflows without forcing the user to repeat conversions by hand.

Worked Example 01

100 mm by 50 mm rectangle

Known

Width (b): 50 mm
Height (h): 100 mm

Formula

Ix = b h³ / 12; Iy = h b³ / 12

Substitution

Ix = 50 × 100³ / 12; Iy = 100 × 50³ / 12

Result

Ix = 4.167×10⁶ mm⁴; Iy = 1.042×10⁶ mm⁴

The same rectangle is four times stiffer about the x-axis than the y-axis because the 100 mm depth is cubed in Ix.

Worked Example 02

200 mm by 300 mm rectangular tube with 10 mm wall

Known

Outer Width (B): 200 mm
Outer Height (H): 300 mm
Wall Thickness (t): 10 mm

Formula

Ix = (B H³ − b h³) / 12; Iy = (H B³ − h b³) / 12

Substitution

Ix = (200×300³ − 180×280³) / 12; Iy = (300×200³ − 280×180³) / 12

Result

Ix = 120.72×10⁶ mm⁴; Iy = 63.92×10⁶ mm⁴; Sx = 804,800 mm³

The inner 180 mm by 280 mm void is subtracted from the outer rectangle. The large outside depth keeps the tube relatively efficient in bending.

Worked Example 03

150 mm outside diameter tube with 10 mm wall

Known

Outer Diameter (D): 150 mm
Wall Thickness (t): 10 mm

Formula

Ix = Iy = π(D⁴ − d⁴) / 64

Substitution

Ix = π × (150⁴ − 130⁴) / 64

Result

Ix = Iy ≈ 10.831×10⁶ mm⁴

A circular tube has equal Ix and Iy about any centroidal diameter. The polar moment for torsion-type checks is J = Ix + Iy for this circular geometry.

Worked Example 04

Symmetric built-up I-section

Known

Overall Depth (H): 300 mm
Flange Width (B): 150 mm
Flange Thickness (t_f): 12 mm
Web Thickness (t_w): 8 mm

Formula

Ix = [B H³ − (B − tw)(H − 2tf)³] / 12; Iy = [2tf B³ + (H − 2tf)tw³] / 12

Substitution

Ix = [150×300³ − (150−8)×276³] / 12; Iy = [2×12×150³ + 276×8³] / 12

Result

Ix ≈ 88.709×10⁶ mm⁴; Iy ≈ 6.762×10⁶ mm⁴

The strong-axis inertia is much larger because the flanges place area far from the horizontal centroidal axis.

Applications

01Finding Ix for beam deflection formulas such as δ = PL³/(48EI)
02Finding section properties before bending stress checks
03Comparing strong-axis and weak-axis stiffness of rectangular, tubular, and I-shaped sections
04Estimating radius of gyration before a column slenderness check

Assumptions

01All formulas use centroidal x and y axes for idealized geometric sections.
02The x-axis is horizontal through the centroid; Ix is usually the strong-axis value when section height is the larger bending dimension.
03Hollow sections use uniform wall thickness and sharp-corner geometry.
04The I-section is symmetric with equal flanges and no fillets or rounded root radii.
05This is area moment of inertia for beam bending, not mass moment of inertia for rotating bodies.

Where This Model Stops

01Does not calculate unsymmetric channel, angle, tee, composite, built-up, or arbitrary polygon section properties.
02Does not apply the parallel-axis theorem for offset axes; values are centroidal only.
03Does not replace manufacturer tables for rolled steel shapes, which include fillets, tapers, tolerances, and standard designations.
04Does not perform beam strength, deflection, buckling, or code-compliance checks by itself.

References

[1]
Second Moment of Area Calculator
Engineer Laboratory
Reference formulas for rectangle, circle, hollow circle, and hollow rectangle second moments of area.
[2]
Pipe and Tubing Formulas
Engineering ToolBox
Pipe/tube formula for moment of inertia I = π(do⁴ − di⁴)/64.
[3]
Moment of Inertia Calculator
EngiToolbox
Explains area moment I versus polar moment J and lists common section formulas.

Frequently Asked Questions

Is area moment of inertia the same as mass moment of inertia?

No. Area moment of inertia, or second moment of area, uses units like mm⁴ and is used for beam bending and deflection. Mass moment of inertia uses units like kg·m² and is used for rotational dynamics.

Should I use Ix or Iy?

Use the value for the bending axis. For a typical vertical beam section bending under gravity loads, Ix is often the strong-axis value because the section depth is vertical. Weak-axis bending usually uses Iy.

Why does depth affect Ix so strongly?

In common formulas, the bending depth is raised to the third or fourth power before division, so moving material farther from the neutral axis increases I very quickly.

Can I use these I-section values for a real W-beam?

Use caution. The I-section here is an idealized sharp-corner built-up section. Standard rolled shapes should be checked against AISC or manufacturer section-property tables.

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