Moment of Inertia Calculator
Calculate second moment of area, section modulus and radius of gyration.
Please note: These are idealised textbook models using representative material data. They are for study, sizing and sanity-checking only — never for final design. Real engineering requires code-compliant analysis, verified material certificates and a qualified engineer’s sign-off.
What the Moment of Inertia Calculator does
Second moment of area measures how a cross-section’s material is distributed relative to the bending axis. It is what makes a joist on edge stiffer than the same joist laid flat, and it feeds directly into every deflection and buckling calculation.
Formula
Rectangle: I = b h³ ÷ 12Circle: I = π d⁴ ÷ 64Section modulus S = I ÷ cRadius of gyration r = √(I ÷ A)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Cross-section | one of 5 options | Yes | — |
| Width | number | In some modes | Shown in 3 of the 5 modes. |
| Width unit | one of 9 options | In some modes | Shown in 3 of the 5 modes. |
| Height / depth | number | In some modes | Shown in 3 of the 5 modes. |
| Height / depth unit | one of 9 options | In some modes | Shown in 3 of the 5 modes. |
| Outer diameter | number | In some modes | Shown Cross-section is Solid circle, or Cross-section is Round tube (hollow circle). |
| Outer diameter unit | one of 9 options | In some modes | Shown Cross-section is Solid circle, or Cross-section is Round tube (hollow circle). |
| Inner diameter | number | In some modes | Shown Cross-section is Round tube (hollow circle). |
| Inner diameter unit | one of 9 options | In some modes | Shown Cross-section is Round tube (hollow circle). |
| Wall thickness | number | In some modes | Shown Cross-section is Rectangular hollow section. |
| Wall thickness unit | one of 9 options | In some modes | Shown Cross-section is Rectangular hollow section. |
| Flange thickness | number | In some modes | Shown Cross-section is I-beam / wide flange. |
| Flange thickness unit | one of 9 options | In some modes | Shown Cross-section is I-beam / wide flange. |
| Web thickness | number | In some modes | Shown Cross-section is I-beam / wide flange. |
| Web thickness unit | one of 9 options | In some modes | Shown Cross-section is I-beam / wide flange. |
| Material | one of 13 options | Yes | — |
How to use it
- Choose Cross-section and Material.
- Fill in the remaining inputs the form shows for your choice.
- Select Calculate.
Worked example
A solid steel rectangle 100 mm wide and 200 mm deep.
- Shape
- Solid rectangle
- Width
- 100 mm
- Height
- 200 mm
I = 100 × 200³ ÷ 12 = 66.67 × 10⁶ mm⁴. S = I ÷ 100 = 666,667 mm³, and r = √(I/A) = 57.7 mm.
Frequently asked questions
What is the difference between I and section modulus?
I governs deflection; section modulus governs stress. S is just I divided by the distance to the outer fibre, so the two carry the same information for different purposes.
Why does laying a joist flat make it so much weaker?
Because b and h swap. A 50 × 200 joist on edge has sixteen times the second moment of area it has lying flat.
Method and sources
Method. Second moment of area for the standard sections offered, computed from the geometric definitions — for a rectangle, I = bh³ ÷ 12.
Assumptions
- The section is as described, homogeneous, and the axis is as stated.
Limitations
- Second moment of *area* governs bending stiffness and is a geometric property. It is distinct from mass moment of inertia, which governs rotation, and the two are routinely confused.
- The axis matters enormously: the same rectangle is far stiffer about one axis than the other, since the depth term is cubed.
- Fabricated sections need the parallel-axis theorem applied about the true centroid, which composite entry does but hand calculation frequently gets wrong.
Sources
- Standard geometric definitions of the second moment of area — Established mechanics of materials. The formulas for each section shape.