Moment of Inertia Calculator
Calculate the second moment of area (moment of inertia) for common cross-sections. Enter dimensions below to get Iyy, Izz, and cross-sectional area instantly.
Moment of Inertia Calculator
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 Izz) and the weak axis (typically y-y, giving Iyy). For symmetric sections like circles and squares, these are equal. For I-beams, Izz is much larger than Iyy—that's why I-beams are oriented with the web vertical.
Formulas Used
Solid Rectangle
Iyy = hb³/12
A = bh
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 Izz by a factor of 8.
Solid Circle
A = πd²/4
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)
Iyy = (HB³ - hb³)/12
A = BH - bh
Subtract the hollow portion from the outer rectangle. B×H are outer dimensions, b×h are inner dimensions.
Hollow Circle (CHS/Pipe)
A = π(D² - d²)/4
D is outer diameter, d is inner diameter. Alternatively, use D and wall thickness t, where d = D - 2t.
I-Beam / H-Section
Iyy = (2tfB³ + hwtw³)/12
A = 2Btf + hwtw
Where hw = H - 2tf is the web height. This simplified formula ignores fillet radii—for precise values, use our full section properties calculator or look up tabulated values.
Why Moment of Inertia Matters
The moment of inertia appears in two fundamental beam equations:
Deflection: δ ∝ 1/(EI)
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 (Iw), and centroid location, use our full structural analysis tool with its comprehensive section library.