Moment of Inertia Calculator

Calculator

Preliminary design aid. Results follow the published formulas cited below and are intended for estimating, study and early design. Final design must be verified by a licensed Professional Engineer against the code in force for your project.

Key Benefits

• ✓Handles both solid and hollow rectangular sections from the same form
• ✓Returns Ixx and Iyy together, so the weak axis is never overlooked
• ✓Also gives area and radius of gyration, the inputs a buckling check needs
• ✓Shows the substitution, so the result can be reproduced by hand
• ✓Sensitivity chart makes the cubic effect of depth immediately visible
• ✓Results export to CSV or share as a link that reproduces your section

What Is Moment of Inertia?

The second moment of area is the integral of y²·dA over a cross-section, where y is the distance from the neutral axis. The square inside that integral is the whole story: material sitting twice as far from the axis contributes four times as much. Carrying out the integration for a rectangle of width b and depth h about its horizontal centroidal axis gives the familiar I = bh³/12, and rotating the section ninety degrees gives I = hb³/12 about the vertical axis.

Why depth dominates

Because h is cubed and b is only linear, depth is the variable worth spending material on. A 100 × 200 mm rectangle standing upright has four times the stiffness of the same rectangle lying flat. This asymmetry explains the shape of nearly every structural member: joists are deep and narrow, I-sections concentrate area in the flanges furthest from the axis, and a hollow tube removes exactly the material nearest the centre where it contributes least.

Hollow sections and subtraction

For a symmetric hollow rectangle the inner void shares the same centroid as the outer shape, so the result is simply the outer moment of inertia minus the inner one: I = (BH³ − bh³)/12. A rectangular hollow section can therefore keep most of its bending stiffness while shedding a large fraction of its weight. When the parts of a built-up shape do not share a centroid, this simple subtraction no longer works and the parallel axis theorem is required.

Formula

Second moment of area of a solid rectangle about its horizontal centroidal axis

How to Use This Calculator

1. Orient the section the way it will be loadedWidth is measured perpendicular to the load and depth parallel to it. Getting this the wrong way round for an upright rectangle understates stiffness by a factor of (h/b)², which for a 100 × 400 section is a factor of sixteen.
2. Enter outer dimensions firstFor a solid section, enter width and height and leave both inner fields at zero. The calculator will return the solid values.
3. Add inner dimensions for a hollow sectionFor a rectangular hollow section, enter the internal clear dimensions. The calculator subtracts the void, which is valid because the outer and inner rectangles share a centroid.
4. Read both axesIxx governs bending about the strong axis, Iyy the weak axis. A column check needs the smaller of the two, since buckling happens in the direction of least resistance.
5. Carry the values into the dependent checksIxx feeds directly into deflection and bending calculations; the radius of gyration and area feed the buckling check. The related calculators below take these numbers as inputs.

Worked Examples

Depth Sensitivity

Because depth is cubed, Ixx rises far faster than area does. Sweep the section depth while holding the width constant to see the two curves diverge — that gap is precisely why deep sections are efficient. The marker shows your current depth.

How to Interpret Your Results

Moment of inertia is a geometric property rather than a pass-or-fail check, so it has no code limit of its own. What matters is the relationship between the numbers: the ratio between the two axes tells you how directional the section is, and the radius of gyration tells you how it will behave in compression.

Common Mistakes to Avoid

Practical Applications

• ▸Computing beam deflection, where I appears directly in the denominator
• ▸Determining bending stress through the section modulus S = I/c
• ▸Supplying I and r to column buckling checks
• ▸Comparing candidate sections by stiffness per unit weight
• ▸Designing built-up and composite members via the parallel axis theorem
• ▸Checking whether a notched or drilled member retains adequate stiffness

Industry Use Cases

Structural steel design

Section tables list I, S, Z and r for every rolled shape, but engineers recompute properties by hand whenever a section is reinforced with cover plates or weakened by a service penetration, since neither case appears in the tables.

Timber engineering

Joists and rafters are usually plain rectangles, so bh³/12 is applied directly. Notching at a bearing reduces the effective depth locally, and because depth is cubed, a notch of a quarter the depth removes well over half the local stiffness.

Product and machine design

Extrusion designers iterate on wall thickness and rib placement to maximise I per unit mass. Because area grows linearly while inertia grows cubically with depth, adding height to a rib is almost always more efficient than thickening it.

Expert Tips

• 💡Doubling depth multiplies stiffness by eight; doubling width only doubles it. Spend material on depth.
• 💡Material near the neutral axis contributes almost nothing — this is exactly what a hollow section exploits.
• 💡The ratio Ixx/Iyy for a rectangle is simply (h/b)², a useful sanity check on any computed pair.
• 💡Keep everything in millimetres. Values in the 10⁶ to 10⁹ mm⁴ range are normal, and a result three orders of magnitude away usually means a unit slip.
• 💡For a built-up section, the A·d² term usually dwarfs the individual Ic terms — compute the offsets carefully.
• 💡A higher radius of gyration at the same area is always the better compression member, whatever the shape.

Advantages & Limitations

Moment of Inertia by Cross-Section Shape

Every shape distributes area differently. The efficiency column compares stiffness per unit area against a solid rectangle of the same depth, which is the fair way to judge whether a shape earns its weight.

Related Calculators

Frequently Asked Questions

Glossary

Second moment of area

The integral of y²·dA over a cross-section, quantifying resistance to bending. Units of length to the fourth power.

Neutral axis

The line through a cross-section where bending stress is zero, passing through the centroid for a homogeneous elastic section.

Centroid

The geometric centre of a cross-section, about which centroidal moments of inertia are measured.

Parallel axis theorem

The relation I = Ic + A·d², used to transfer a part's moment of inertia to an axis offset by distance d.

Radius of gyration

The quantity √(I/A), representing the distance at which the whole area could be concentrated to give the same moment of inertia.

Principal axes

The pair of perpendicular axes through the centroid about which the product of inertia vanishes, giving the maximum and minimum values of I.

Section modulus

I divided by the distance to the extreme fibre, S = I/c, used for bending stress rather than stiffness.

Flexural rigidity

The product EI of modulus of elasticity and moment of inertia — the quantity that actually governs deflection.

Rectangular hollow section

A closed rectangular tube whose inner void shares the outer centroid, allowing its inertia to be found by simple subtraction.

Scientific & Standards References

1. AISC Steel Construction Manual, 16th Edition — Part 1: Dimensions and Properties — American Institute of Steel Construction
2. Gere, J. M. & Goodno, B. J., Mechanics of Materials, 9th Edition — Appendix D: Properties of Plane Areas — Cengage Learning
3. Hibbeler, R. C., Structural Analysis, 10th Edition — Appendix A: Geometric Properties of an Area — Pearson
4. EN 1993-1-1 §6.2 — Resistance of cross-sections — CEN
5. Roark's Formulas for Stress and Strain, 9th Edition — Table A.1: Properties of Sections — McGraw-Hill

Conclusion

The second moment of area is the geometric property behind every bending and buckling check, and for a rectangle it reduces to I = bh³/12. The cubic dependence on depth is the practical lesson: depth buys stiffness far more cheaply than width, material near the neutral axis earns almost nothing, and removing that core is why hollow sections outperform solid ones per unit weight. Compute both axes as a matter of routine — the weak-axis value is the one a column check needs, and it is the one most often overlooked.