Moment of Inertia Calculator

Calculate the second moment of area (moment of inertia) for common cross-sections. Enter dimensions below to get I_yy, I_zz, and cross-sectional area instantly.

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What is Moment of Inertia?

The moment of inertia (more precisely, the second moment of area) measures how a cross-section's area is distributed relative to an axis. For the same material, span, supports, and loading, a larger second moment of area produces a stiffer beam.

Engineers use two main centroidal axes through the section. The strong axis is the one with the larger second moment of area, and the weak axis has the smaller value. For circles and squares, these are equal. For a typical deep I-beam, I_zz is much larger than I_yy-one reason the web is commonly oriented vertically for gravity bending.

Formulas Used

Solid Rectangle

Where b is width and h is height. The second moment of area increases with the cube of the dimension perpendicular to the bending axis; doubling the height increases I_zz by a factor of eight.

Solid Circle

For a circle, both axes have equal moment of inertia. The formula can also be written as I = πr⁴/4 using the radius.

Hollow Rectangle (RHS/SHS)

Subtract the hollow portion from the outer rectangle. B×H are outer dimensions, b×h are inner dimensions.

Hollow Circle (CHS/Pipe)

D is outer diameter, d is inner diameter. Alternatively, use D and wall thickness t, where d = D - 2t.

I-Beam / H-Section

Where h_w = H - 2t_f is the web height. This idealised formula ignores fillet radii; for catalogue sections, compare against the supplier's tabulated values.

Channel (C-Section)

A channel has the same I_zz formula as an I-beam (both are symmetric about the strong axis). However, I_yy requires finding the centroid position from the web back and applying the parallel axis theorem, since the flanges extend to one side only.

Tee (T-Section)

A tee section is not symmetric about the strong axis, so the centroid must be calculated first. I_zz then uses the parallel axis theorem for both the web and flange. I_yy is straightforward since the section is symmetric about the vertical axis.

Angle (L-Section)

Angles are not symmetric about either geometric axis, so both the centroid and the parallel-axis theorem are needed. These results are about centroidal axes parallel to the legs. Buckling calculations may require rotated principal-axis properties; use the section properties calculator for those.

Why Moment of Inertia Matters

The moment of inertia appears in two fundamental beam equations:

For a given extreme-fibre distance, higher I means lower elastic bending stress for the same moment and less elastic deflection for the same loading. Efficient sections place material away from the relevant neutral axis to increase I for a given area.

Need More Section Properties?

This page reports area and centroidal second moments of area for idealised dimensions. Use the section properties calculator for centroid, elastic and plastic section modulus, torsion constant, and the warping properties implemented for supported shapes.

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