How to use the Moment of Inertia Calculator
- 1. Pick the shapeRectangle, hollow box, circle, tube, triangle, I, T or channel.
- 2. Enter sizes in mmFlange widths, thicknesses, depth and web thickness.
- 3. Read the propertiesCentroid, Ixx, Iyy, section modulus, radius of gyration and the working.
Formula and method
For a rectangle about its own centroid, I = bh³/12. For built-up sections, split the shape into rectangles, find the centroid with ȳ = ΣAy / ΣA, then use the parallel axis theorem: I = Σ(Iown + A·d²), where d is the distance from each part's centroid to the section centroid.
Section modulus Z = I / ymax gives bending stress (σ = M/Z), and radius of gyration k = √(I/A) is used in column buckling.
Worked example: T-section
Flange 100 × 20 mm on a web 20 × 80 mm (overall depth 100 mm). Areas: flange 2000 mm², web 1600 mm². ȳ from bottom = (2000 × 90 + 1600 × 40) / 3600 = 67.78 mm.
Ixx = 100×20³/12 + 2000 × 22.22² + 20×80³/12 + 1600 × 27.78² = 3.142 × 10⁶ mm⁴.
- Exam tip: always measure y from one reference edge, usually the bottom.
- Holes are treated as negative areas.
- Check that ȳ lies inside the section; if not, recheck the areas.
Frequently asked questions
What is the difference between Ixx and Iyy?
Ixx is about the horizontal axis through the centroid and controls bending under vertical loads. Iyy is about the vertical axis and controls sideways bending and buckling about the weak axis.
Why is polar moment shown as Ixx + Iyy?
That is the polar moment about the centroid. It equals the torsion constant only for circular sections. Thin open sections need a different torsion constant.
Can I use cm instead of mm?
Enter sizes in mm. The result is shown in mm⁴ and also in cm⁴ (divide by 10⁴), which is what steel tables use.
Is this calculator free?
Yes. You enter your name and WhatsApp number once to see your result, and all other Uncodemy tools then open without the form. Everything is calculated in your browser.