Moment of Inertia for Standard Cross-Sections
Calculate second moment of area (moment of inertia), section modulus, and radius of gyration for rectangle, circle, I-beam, T-section, and more.
The moment of inertia (or second moment of area) of a cross-section measures how resistant that shape is to bending about a given axis. A deeper beam is exponentially stiffer than a wider one because the formula cubes the height dimension. This is why I-beams are shaped the way they are — material is placed as far as possible from the neutral axis, where it does the most work resisting bending.
Symbol: I (mm⁴ or cm⁴ or in⁴) Key formula for bending stress: σ = M × y / I = M / Z
Where:
- M = bending moment (N·mm or lb·in)
- y = distance from neutral axis to extreme fiber
- Z = I / y_max = Section Modulus (mm³): what engineers actually use
Formulas by Section Type:
| Section | I_x (strong axis) | Area |
|---|---|---|
| Solid Rectangle | b×h³/12 | b×h |
| Hollow Rectangle | (b×h³ − (b−2t)×(h−2t)³)/12 | b×h − (b−2t)×(h−2t) |
| Solid Circle (d=diameter) | π×d⁴/64 | π×d²/4 |
| Hollow Circle (D=outer, d=inner) | π×(D⁴−d⁴)/64 | π×(D²−d²)/4 |
| I-Beam (bf=flange width, H=total height, tf=flange thick, tw=web thick) | bf×H³/12 − (bf−tw)×hw³/12 | bf×2tf + hw×tw |
Where hw = H − 2×tf for I-beams.
Section Modulus Z = I / y_max For a rectangle: y_max = h/2, so Z = b×h²/6 For a circle: y_max = d/2, so Z = π×d³/32
Radius of Gyration r = √(I/A) Used in column buckling. The Euler critical load formula is: P_cr = π²×E×I / (K×L)² Where K×L is the effective column length. A higher r means a column can be longer before buckling.
Real Example: W8×31 Steel I-Beam (AISC): Actual properties: Ix = 110 in⁴, Sx = 27.5 in³, rx = 3.47 in If a beam carries M = 50 kip·ft = 600 kip·in: Bending stress σ = M/S = 600/27.5 = 21.8 ksi (well under A36 yield of 36 ksi)
Why Section Shape Matters: For the same cross-sectional area (same amount of steel), an I-beam has roughly 3–5× the moment of inertia of a solid rectangle. This is why wide-flange beams dominate structural steel construction — they are far more efficient than square bars.
Units Tip: Use consistent units throughout. If dimensions are in mm, I will be in mm⁴, and stress will be in MPa when moment is in N·mm.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.