Free 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.

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. It determines how much a beam will bend under load-the higher the moment of inertia, the stiffer the beam.

Engineers use two main axes: the strong axis (typically z-z, giving I_zz) and the weak axis (typically y-y, giving I_yy). For symmetric sections like circles and squares, these are equal. For I-beams, I_zz is much larger than I_yy-that's why I-beams are oriented with the web vertical.

Formulas Used

Solid Rectangle

Where b is width and h is height. The moment of inertia increases with the cube of the dimension perpendicular to the bending axis-doubling the height increases I_zz by a factor of 8.

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 simplified formula ignores fillet radii-for precise values, use our full section properties calculator or look up 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 axis, so both the centroid and the parallel axis theorem are needed for both I_zz and I_yy. Note these are geometric-axis properties. For buckling calculations, principal-axis properties (rotated) may be required-use the section properties calculator for those.

Why Moment of Inertia Matters

The moment of inertia appears in two fundamental beam equations:

Higher I means lower stress for the same moment, and less deflection for the same load. This is why efficient structural sections (I-beams, channels, hollow sections) concentrate material away from the neutral axis-maximising I while minimising weight.

Need More Section Properties?

This calculator provides the essential properties: area and moment of inertia. For complete section analysis including torsion constant (J), section modulus (S), warping constant (I_w), and centroid location, use our full structural analysis tool with its comprehensive section library.

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