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Cross-Section Properties Calculator

Free cross-section calculator: area, centroid, moment of inertia (Ix, Iy), section modulus, radius of gyration & plastic modulus for I-beams, pipes and custom polygons.

Pick a standard structural shape or draw a custom polygon to get exact geometric properties — area, centroid, moments of inertia, section moduli, radii of gyration, polar and principal moments, and plastic moduli. Everything is computed in your browser; nothing is uploaded.
mm
mm
── Neutral axes   ⨁ Centroid

What is a cross-section properties calculator?

A cross-section (or section) is the 2D shape you see when you slice through a beam, column or any structural member. Its geometric properties — area, centroid, moment of inertia, section modulus and radius of gyration — govern how stiff and strong the member is in bending, buckling and torsion. This tool computes all of them exactly using Green's theorem for polygons and closed-form formulas for circular shapes, then draws the section in a clean blueprint style with the centroid and neutral axes marked.

How to use it

  1. Choose a standard shape (rectangle, hollow box, circle, pipe, I-beam, T, angle or channel) — or pick Custom polygon.
  2. Select your units (mm, cm, m, in or ft). All dimensions and results use that unit.
  3. Enter the dimensions, or for a custom polygon click on the grid / paste x,y coordinates.
  4. Press Calculate to get the full property table and the blueprint diagram.

Properties computed

  • A — Cross-sectional area.
  • x̄, ȳ — Centroid coordinates of the section.
  • Ix, Iy — Second moments of area about the centroidal x and y axes — the key inputs to bending and deflection.
  • Ixy — Product of inertia (zero for symmetric sections).
  • Sx, Sy — Elastic section moduli, I divided by the extreme-fibre distance — used for bending stress.
  • rx, ry — Radii of gyration, √(I/A) — used in column buckling (slenderness).
  • Ip — Polar moment of inertia about the centroid, Ix + Iy.
  • Zx, Zy — Plastic section moduli — used in plastic / limit-state design.
  • I₁, I₂, θ — Principal moments of inertia and the principal-axis angle.

Formulas

Area (shoelace / Green's theorem):

A = ½ Σ (xi yi+1 − xi+1 yi)

Second moment of area:

Ix = 1⁄12 Σ (yi² + yiyi+1 + yi+1²)(xi yi+1 − xi+1 yi)

Section modulus and radius of gyration:

Sx = Ix ⁄ c    rx = √(Ix ⁄ A)

Frequently Asked Questions

For polygons the moments are computed analytically with Green's theorem, so they are exact (no meshing or approximation). Circular shapes use closed-form formulas. Results are limited only by the precision of your input dimensions.

Only for solid and hollow circular sections. For all other shapes Ip = Ix + Iy is the polar moment of inertia, which is generally larger than the St. Venant torsion constant J. This tool reports Ip, not J.

Yes. Choose Custom polygon and click on the grid to place vertices, or paste a list of x,y coordinates. The tool computes area, centroid, inertia, section and plastic moduli for any simple (non-self-intersecting) polygon.

Plastic modulus is computed for single solid outlines. For hollow polygonal sections the plastic neutral axis split is omitted to avoid misleading values; elastic properties (Ix, Sx, etc.) are still exact.

No. All math runs locally in JavaScript. Nothing is uploaded, so it works offline and keeps your engineering data private.
Cross-Section Properties Calculator — Free cross-section calculator: area, centroid, moment of inertia (Ix, Iy), section modulus, radius of gyration & plastic
Cross-Section Properties Calculator
See also
Engineering CalculatorsENGINEERING CALCULATORS40
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