Composite Section Moment of Inertia Calculator (Centroid, Ix, Iy, Section Modulus)

See Composite Section Moment of Inertia Calculator (Centroid, Ix, Iy, Section Modulus) in action

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1Add shapes and enter their size and bounding-box centre.

2Tick Hole for cut-outs and rotate as needed.

3Read the section properties.

Free Composite Section Moment of Inertia Calculator

Real cross-sections are rarely a single standard shape. This calculator lets you build one from simple pieces — rectangles, circles, ellipses, isosceles and right triangles and semicircles, any of which can be a hole — place each piece by the coordinates of its bounding-box centre, rotate it in 90° steps, and get the properties of the whole section: area, centroid, centroidal moments of inertia Ix and Iy, the product of inertia, the polar moment, principal moments and angle, section moduli at the extreme fibres and radii of gyration. A live drawing shows the shapes, holes in dashed red, and the centroid with its axes.

Each piece's own centroidal inertia is transferred to the combined centroid with the parallel-axis theorem and holes are subtracted. Examples load a T-section, an I-beam, a round tube, a rectangle with a semicircular cap and an L-section with a hole. Results were verified against closed-form solutions: circle πd⁴/64, semicircle (π/8 − 8/9π)r⁴ about its centroid and a T-section by parallel axis.

Key features

How to use it

1. Add shapes and enter their size and bounding-box centre.
2. Tick Hole for cut-outs and rotate as needed.
3. Read the section properties.
4. Compare with the drawing to check the layout.

Worked example

Example

A T-section built from a 100 × 20 mm flange on top of a 20 × 80 mm web has an area of 3,600 mm², a centroid 67.8 mm above the bottom and Ix of about 3.14 × 10⁶ mm⁴.

Who uses this tool

Tips for the best results

• Give positions as the centre of each shape's bounding box.
• Use holes instead of subtracting by hand.
• Check that the drawing matches your intent before trusting numbers.
• Use the section modulus S = I/c for stress: σ = M/S.

Common mistakes to avoid

• Entering the corner position instead of the centre.
• Forgetting to mark a cut-out as a hole.
• Using Iy when you need Ix — Ix is about the horizontal axis.

Frequently asked questions