What this calculator does
The second moment of area, also called the area moment of inertia, measures how a cross-section's material is distributed relative to a bending axis. A larger second moment of area means the shape resists bending more effectively for the same cross-sectional area, which is why it is central to beam and structural design.
This second moment of area calculator covers the three standard shapes with well-known closed-form formulas: a solid rectangle (I = bh³/12), a solid circle (I = πd⁴/64) and a hollow circular tube (I = π(D⁴−d⁴)/64). For circular sections it also gives the polar moment of area (J), which describes resistance to twisting rather than bending, and for a solid or hollow circle is always exactly twice the ordinary second moment of area.
The formula
For a rectangle, multiply the width by the cube of the height and divide by 12. For a solid circle, take π times the diameter to the fourth power, divided by 64; the polar moment of area for the same circle is that figure divided by 32 instead (exactly double the ordinary second moment). For a hollow tube, subtract the inner diameter to the fourth power from the outer diameter to the fourth power before applying the same circle formulas.
| Term | Meaning |
|---|---|
| I | Second moment of area (area moment of inertia): resistance to bending about a given axis through the centroid. |
| J | Polar second moment of area: resistance to twisting (torsion) about an axis through the centroid, perpendicular to the cross-section. |
| b, h | Width and height of a rectangular cross-section. |
| d, D | Diameter of a solid circle, or the outer diameter of a hollow tube. |
| Z | Section modulus: I divided by the distance from the centroid to the extreme fibre, used to relate bending moment to stress. |
The inputs explained
| Field | What to enter |
|---|---|
| Cross-section | Choose the cross-section shape: a solid rectangle, a solid circle, or a hollow circular tube. |
| Width (b), rectangle only (mm) | The width of the rectangle, only used when the rectangle shape is selected. |
| Height (h), rectangle only (mm) | The height of the rectangle, measured in the direction the bending axis runs across, only used for the rectangle shape. |
| Outer diameter (d or D) (mm) | The diameter of a solid circle, or the outer diameter of a hollow tube. |
| Inner diameter, tube only (mm) | The inner diameter of a hollow tube, only used for the tube shape; must be smaller than the outer diameter. |
When to use it
Sizing a rectangular beam
Second moment of area for a rectangular timber or steel section feeds directly into beam deflection and bending-stress calculations, and it grows with the cube of the height, which is why standing a beam on edge is far stiffer than laying it flat.
Checking a round shaft or column
Solid circular sections are common for shafts and columns; both the bending (I) and torsional (J) second moments of area are needed depending on whether the shaft is being bent or twisted.
Working with a hollow tube section
Tubular sections, common in scaffolding and bicycle frames, achieve a high second moment of area relative to their weight by moving material away from the centre; the hollow tube formula accounts for that removed inner material.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How second moment of area changes with height for a fixed-width rectangle
A fixed 50 mm width, across a range of section heights.
| Height | Second moment of area (I) | Section modulus (Z = I ÷ h/2) |
|---|---|---|
| 50 mm | 520,833.33 mm⁴ | 20,833.33 mm³ |
| 75 mm | 1,757,812.50 mm⁴ | 46,875.00 mm³ |
| 100 mm | 4,166,666.67 mm⁴ | 83,333.33 mm³ |
| 150 mm | 14,062,500.00 mm⁴ | 187,500.00 mm³ |
| 200 mm | 33,333,333.33 mm⁴ | 333,333.33 mm³ |
| 300 mm | 112,500,000.00 mm⁴ | 750,000.00 mm³ |
How second moment and polar moment of area change with diameter for a solid circle
A range of solid circle diameters.
| Diameter | Second moment of area (I) | Polar second moment of area (J) |
|---|---|---|
| 20 mm | 7,853.98 mm⁴ | 15,707.96 mm⁴ |
| 30 mm | 39,760.78 mm⁴ | 79,521.56 mm⁴ |
| 50 mm | 306,796.16 mm⁴ | 613,592.32 mm⁴ |
| 75 mm | 1,553,155.55 mm⁴ | 3,106,311.10 mm⁴ |
| 100 mm | 4,908,738.52 mm⁴ | 9,817,477.04 mm⁴ |
| 150 mm | 24,850,488.76 mm⁴ | 49,700,977.53 mm⁴ |
Questions
What is the difference between second moment of area and polar moment of area?
The ordinary second moment of area (I) resists bending about an axis lying in the plane of the cross-section. The polar moment of area (J) resists twisting about an axis perpendicular to the cross-section. For a circle or tube, J is always exactly twice I, but this relationship does not hold for non-circular shapes.
Why does height matter so much more than width for a rectangular beam?
The rectangle formula is I = bh³/12: height is cubed but width is not. Doubling a beam's height increases its bending stiffness eightfold, while doubling its width only doubles it, which is why beams are almost always oriented with their greater dimension vertical.
Is second moment of area the same as mass moment of inertia?
No, despite the similar name. Second moment of area is a purely geometric property of a cross-section, used in bending and torsion calculations. Mass moment of inertia describes how an object's mass is distributed for rotational dynamics, and has entirely different units.
How does this feed into beam deflection?
Beam deflection depends on the product EI, where E is the material's elastic modulus and I is the second moment of area of its cross-section. This calculator provides the I value, which then plugs into the beam deflection calculator alongside the material and load.
Once you have the second moment of area, use it in the beam deflection calculator to find how much a beam bends under load. For a related deformation measure, see the shear strain calculator.