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.
| 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 |
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.
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.
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.
This tool computes the area moment of inertia, the second moment of area of a cross-section about an axis, with units of length to the fourth power. It is the I in the bending stress formula and in beam deflection formulas. Mass moment of inertia is a different quantity, has units of mass times length squared, and describes resistance to angular acceleration in dynamics.
Draw your section anywhere on the canvas. The centroid is reported in the canvas coordinates you drew in, but Ix, Iy, Ixy, J and the section moduli are all reported about the centroidal axes, so where you place the section makes no difference to them.
Draw the outer shape first, then draw the hole as a second rectangle and toggle it to Void in the shape list. The void subtracts both its area and its parallel axis term. An I-beam is three solid rectangles, a hollow box is one rectangle with one void, and a C-channel works either way.
Section modulus is I divided by the distance from the centroid to the fiber in question. For a section that is not symmetric about the bending axis, such as a T-section, the centroid sits closer to one face, so the two distances differ and so do the two moduli. Design against the smaller one, because that fiber sees the higher stress.