Moment of Inertia & Section Properties Calculator
Every section property of a standard or custom shape, with the composite-area working.
Preview
All section properties
| Property | mm | cm |
|---|
Parts (composite-area method)
| Part | A | x̄ | ȳ | Ix share | Iy share |
|---|
Each share is the part’s own second moment plus A·d² (parallel-axis theorem); holes count negative. The shares add up to Ix and Iy.
How it was calculated
Every axis
| Axis | I | k | I + m·d² |
|---|
k is the radius of gyration, √(I ÷ m): the radius at which all the mass would give the same inertia.
Formulas
About the Moment of Inertia & Section Properties Calculator
Get the section properties of a cross-section: area, centroid, second moments of area I_x and I_y, product of inertia I_xy, polar moment J, principal moments and their angle, elastic section moduli for every extreme fibre, plastic section moduli with the plastic neutral axis, and radii of gyration. Choose a rectangle, hollow rectangle (with EN 10219 or EN 10210 corner radii), circle, tube, I- or H-section (equal or unequal flanges), T, channel, angle or triangle — rolled shapes can include their root radius — or build your own from rectangles and circles, or paste polygon coordinates with holes.
A to-scale preview shows the centroid, the axes and the plastic neutral axis, and a composite-area table lists each part’s share of I by the parallel-axis theorem, so you can check the working by hand. A second mode gives the mass moment of inertia of cylinders, tubes, spheres, blocks, cones, rods and rings from their size and mass or density.
How to use it
- Choose Section properties or Mass moment of inertia.
- For a section, pick the shape and the unit, then enter the sizes. For an I-section untick Same flanges to give the top and bottom flanges their own sizes. Rolled shapes take the root radius from the steel table.
- For a custom shape choose Composite and add rectangles and circles (solid or hole), or Polygon and type or paste one “x, y” point per line; a line saying
holestarts an outline to cut out. - Check the preview, then read the properties. The table gives each value in your unit and a second one (mm and cm, for example); download it as CSV or copy it.
- For a solid, pick the shape and enter its size and either its mass or its material density. Add a distance d to move the axis with the parallel-axis theorem.
Examples
A = 20,000 mm², I_x = bh³/12 = 66.67 × 10⁶ mm⁴, S_x = bh²/6 = 666,667 mm³, Z_x = bh²/4 = 1.0 × 10⁶ mm³, r_x = 57.74 mm
A = 53.8 cm², I_y = 8,356 cm⁴, W_el = 557.1 cm³, W_pl = 628.4 cm³, I_z = 603.8 cm⁴ — the published values (Euro tables call the strong axis y)
A = 18.4 cm², I = 271 cm⁴, W_el = 54.2 cm³, W_pl = 64.6 cm³, i = 3.84 cm
I_x = I_y = 180.0 cm⁴, I_xy = −106.6 cm⁴, principal I₁ = 286.6 cm⁴ and I₂ = 73.4 cm⁴ at 45°
Density 7,850 kg/m³
m = 123.3 kg, I about its axis = mr²/2 = 0.617 kg·m², about a diameter through the centre = 2.877 kg·m²
Common uses
- Getting I and S for a beam-deflection or stress check
- Checking a fabricated or built-up section (plate girder, box, channel with a plate) against its parts
- Section properties of an extrusion or cut-out traced in CAD, pasted as coordinates
- Finding the rotational inertia of a flywheel, shaft or rotor for motor sizing
Formulas
- Centroid: x̄ = ΣAᵢxᵢ ÷ ΣAᵢ, ȳ = ΣAᵢyᵢ ÷ ΣAᵢ (holes count negative)
- Parallel-axis theorem: I_x = Σ(I_x,i + Aᵢ dᵢ²) about the centroid of the whole section
- Polygons (shoelace): A = ½ Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ), I_x = 1/12 Σ(yᵢ² + yᵢyᵢ₊₁ + yᵢ₊₁²)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)
- Principal axes: I₁,₂ = (I_x + I_y)/2 ± √[((I_x − I_y)/2)² + I_xy²], tan 2θ = −2I_xy ÷ (I_x − I_y)
- Elastic modulus: S = I ÷ c, with c the distance to the extreme fibre (top and bottom differ for unsymmetric shapes)
- Plastic modulus: Z = A_c·ȳ_c + A_t·ȳ_t about the axis that splits the area in half
- Radius of gyration: r = √(I ÷ A); polar moment: J = I_x + I_y
Which axis is which
Here x is horizontal and y vertical as drawn, so I_x is the strong-axis value of a beam drawn upright. European tables (EN 1993) call that axis y–y and the weak axis z–z; American tables (AISC) use x–x and y–y like this calculator. Angles and other unsymmetric shapes bend about their principal axes 1 and 2 (u–u and v–v in tables) unless they are held, so check I₂ and r_min for buckling.
Torsion is different
J here is the polar moment of area, I_x + I_y. It equals the torsion constant only for solid and hollow circles. Open sections such as I-beams and channels are far weaker in torsion — roughly Σ b t³ ÷ 3 — so use the torsion constant from the steel table for twisting.
Sources
- Beer, Johnston, DeWolf & Mazurek, Mechanics of Materials, 8th ed. (2020), appendices on area and mass moments of inertia
- AISC Steel Construction Manual, 16th ed., Part 17 (properties of geometric sections)
- EN 10219-2 and EN 10210-2 (hollow-section corner radii for calculating properties)
- Polygon section properties from Green’s theorem (the shoelace formulas)
Limitations
- Taper flanges, toe radii and lips of cold-formed sections are not modelled; rounded corners and root fillets are drawn with short straight chords, which changes the results by far less than rolling tolerances.
- Composite parts must touch without overlapping and holes must lie inside a solid part — the tool warns when they do not. Different materials in one section (transformed sections) are not handled.
- Pasted outlines can have up to 20,000 points in all; an outline of more than 2,000 points is not checked for crossing edges (the page says so), so its points must run in order around it.
- The polar moment J is not the torsion constant of non-circular sections, and warping constants are not given.
- Mass moments of inertia assume uniform density and ideal shapes; the thin rod and ring use their mass only.
Privacy
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Frequently asked questions
What is the moment of inertia of a rectangle?
About the horizontal axis through its centroid, I_x = bh³ ÷ 12, with b the width and h the height (the dimension in the direction of bending). A 100 × 200 mm rectangle gives 66.7 × 10⁶ mm⁴ standing up but only 16.7 × 10⁶ mm⁴ lying flat, which is why joists are placed on edge.
What is the difference between the elastic and plastic section modulus?
The elastic modulus S = I ÷ c gives the moment at which the extreme fibre first yields (M = f_y S), and the bending stress σ = M ÷ S used in the beam deflection calculator. The plastic modulus Z gives the moment when the whole section has yielded (M = f_y Z). Their ratio, the shape factor, is 1.5 for a rectangle and about 1.1–1.2 for I-beams.
How do I find the centroid of a composite shape?
Split it into simple parts, multiply each part’s area by the position of its centroid, add them up and divide by the total area. The parts table here shows every part’s area and centroid, so you can follow it step by step.
How does the parallel-axis theorem work?
The second moment about any axis equals the second moment about the parallel axis through the centroid plus the area times the distance between the axes squared: I = I_c + A·d². For mass moments of inertia it is I = I_c + m·d².
Why are my results slightly different from the steel table?
Tables include the root radius and, for some shapes, tapered flanges and rounded toes. Enter the root radius for I-sections, channels, T-sections and angles, and the EN 10219 or EN 10210 corners for hollow sections; the IPE 300 and SHS examples above then agree with the tables.
What units should I use?
Any — enter every size in the unit you choose and the results come out in that unit raised to the right power (mm², mm⁴, mm³). The table also converts each value to a second unit, such as cm⁴ and cm³ used in European tables.