Moment of Inertia and Centroid Calculator

Section Properties Calculator

This tool calculates section properties for composite cross-sections made of rectangles and rectangular voids. Draw shapes on the canvas, use the Dimension tool to set exact widths and heights, then read off centroid coordinates, area moments of inertia, section moduli, product of inertia, and polar moment directly from the right panel. No formulas to look up. No spreadsheet to maintain. Nothing leaves your browser.

How to use the section properties tool

1. Sketch your section. Click and drag on the canvas with the Draw tool to place rectangles. Position them roughly, overlap is fine at this stage.
2. Set exact dimensions. Switch to the Dimension tool and click any shape to edit its width, height, and position numerically. The tool solves linked dimensions simultaneously so constrained shapes snap into place.
3. Mark voids. Select any rectangle in the shape list on the left and toggle it to Void. The tool subtracts that area from the composite section using the parallel axis theorem.
4. Read results. The right panel updates live. Export to PNG (paste directly into a Word or Google Docs report), download SVG, or save a PDF.

Section properties computed

Property	Symbol	Description
Total area	A	Sum of all solid regions minus voids
Centroid	x̅, y̅	Area-weighted average of sub-shape centroids
Area moment of inertia	Ix, Iy	Second moment of area about centroidal axes
Product of inertia	Ixy	Cross-product of area about centroidal axes
Polar moment of inertia	J	Ix + Iy, used for torsion of non-circular sections
Section modulus	Sx, Sy	I divided by distance to extreme fiber; relates bending stress to moment

Parallel axis theorem

The calculator applies the parallel axis theorem to each sub-region automatically. For a rectangle of area A with its own centroidal moment I₀, the contribution to the composite Ix is:

I₀ + A·d², where d is the vertical distance from that sub-shape's centroid to the composite centroid.

Void regions subtract their area and their parallel axis term. This is the standard approach for I-beams, T-sections, C-channels, hollow rectangles, and any other built-up section you can compose from rectangles.

Common cross-section types

• I-beam (W-shape or S-shape): Three rectangles -- top flange, web, bottom flange.
• T-section: Flange rectangle stacked on a web rectangle.
• C-channel: Three rectangles arranged in a C, or one large rectangle with a void cut from the open side.
• Hollow rectangle: One outer rectangle with a centered void.
• L-angle: Two rectangles sharing a corner.

For circular or curved sections, Roark's Formulas for Stress and Strain (8th Ed.) provides closed-form results that can be entered as equivalent rectangular regions for combined sections.

Using results in structural and machine design

The bending stress formula σ = M·c / I requires Ix and the distance from the centroid to the extreme fiber. The section modulus Sx = Ix / c collapses these into one value, which is why beam tables list Sx directly. For shaft torsion, the polar moment J appears in τ = T·r / J. This tool gives you both.

Results from this calculator match what you would find in AISC steel section tables for rectangular cross-sections, and serve as a check against textbook worked examples in Hibbeler's Mechanics of Materials and Beer and Johnston's Mechanics of Materials.

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