Moment of Inertia Calculator guide
Moment of Inertia Calculator: Second Moment of Area for Beams
Use our moment of inertia calculator to find the second moment of area, section modulus and radius of gyration for rectangles, tubes, I-beams and more.
By the AllCalculator editorial teamUpdated 15 October 20262 min read
How much a beam bends depends on its material and on the shape of its cross-section. The second moment of area, often called the area moment of inertia, measures how well a shape resists bending. Our Moment of Inertia Calculator works it out for common sections.
For structural work, always use a qualified engineer and the manufacturer's published section properties.
How to use the calculator (Moment of Inertia Calculator)
- Choose the cross-section: rectangle, box section, round bar, tube, I-beam or triangle.
- Enter the dimensions in millimetres.
You'll see Ix and Iy, the area, the elastic section modulus and the radius of gyration.
The formulas
| Section | Ix (about the horizontal axis) |
|---|---|
| Rectangle b × h | b·h³ / 12 |
| Round bar, diameter d | π·d⁴ / 64 |
| Tube, outside d, inside dᵢ | π·(d⁴ − dᵢ⁴) / 64 |
| Triangle (centroid) | b·h³ / 36 |
Why depth matters so much
Because h is cubed, doubling a beam's depth makes it eight times stiffer, while doubling its width only doubles the stiffness. That's why joists stand on edge and I-beams put most of their material in the flanges, far from the centre.
A worked example
With the calculator's default settings, a 100mm × 200mm rectangle:
- Ix = 100 × 200³ ÷ 12 = 66,666,667mm⁴, or 6,667cm⁴
- Section modulus Zx = Ix ÷ 100 = 667cm³
Turn it on its side and Iy is only 1,667cm⁴, a quarter of the stiffness.
Section modulus and radius of gyration
- Section modulus (Z = I ÷ c) links bending moment to stress: stress = M ÷ Z.
- Radius of gyration (r = √(I ÷ A)) is used in buckling calculations for columns.
For working with areas and volumes, see our Square Meter Calculator and Concrete Calculator.
Frequently asked questions
Is this the same as mass moment of inertia?
No. This is the second moment of area, used for bending. Mass moment of inertia is about rotation and depends on mass.
What units are used?
mm⁴ and cm⁴ for I, and cm³ for Z. 1cm⁴ is 10,000mm⁴.
Can I use it for steel sections?
For a quick check, yes. For design, use the published section tables, which include root radii and exact dimensions.
Use the Moment of Inertia Calculator to compare sections and understand how shape affects stiffness.
Sources and further reading
- The Institution of Structural Engineersistructe.org
- SCI Steel Construction: Section propertiessteelconstruction.info
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.