Structural · Section properties
Moment of Inertia Calculator
Compute area I and polar J for seven standard profiles — the cross-section diagram highlights whichever dimension you are editing.
Cross-section profile
Solid rectangle — classic beam and plate section
Dimensions
Section preview
Scales to your entered dimensions · highlight follows the active field
Result
Area moment of inertia
I = bh³/12
Choose area I for bending beams, polar J for torsion — the live diagram tracks whichever dimension you edit.
Second moment of area, polar J, and the formulas behind each profile in this calculator.
The area moment of inertia (second moment of area, I) measures a cross-section's resistance to bending. Beams with larger I deflect less under the same load and material. Formulas in this calculator use centroidal axes unless noted for asymmetric profiles.
- RectangleI = bh³/12
- CircleI = πd⁴/64
- I-beamI = (bf·h³ − (bf−tw)(h−2tf)³)/12
- Hollow rectangleI = (BH³ − bh³)/12
- Hollow circleI = π(D⁴ − d⁴)/64
- L-angle / channelIₓ, I_y, I_xy → principal I₁, I₂
Area moment I
- Transverse bending and beam deflection
- σ = My/I, δ = PL³/(3EI)
- Floor joists, bridge girders, columns in bending
Polar moment J
- Torsion and twist of shafts
- τ = Tr/J, θ = TL/(GJ)
- Drive shafts, propellers, drill strings
J = Iₓ + I_y · solid circle: J = 2I
- All internal calculations use millimetres; convert result units as needed.
- Asymmetric sections (L-angle, channel) report principal axes I₁, I₂ and angle θ.
- For composite shapes, apply the parallel axis theorem to each part.
- Units are length⁴ — mm⁴, cm⁴, in⁴, etc.
- An I-beam's flanges contribute most of its bending stiffness — the web mainly carries shear.
- For a solid circle, polar J is exactly twice the area moment I — a useful sanity check.
- Moving material outward (hollow sections) boosts I and J without adding much weight.
- The parallel axis theorem lets you shift I to any axis: I = I_c + Ad².
- Steel W-shapes in AISC manuals list I about the strong axis — match your calculator axis.
- Composite sections sum individual I values after shifting each part to the neutral axis.
Worked example: I for a 50 × 100 mm rectangle about the 100 mm axis
- I = b × h³ / 12
- I = 50 × 100³ / 12
I ≈ 4.17 × 10⁶ mm⁴
What is the second moment of area?
The second moment of area (I) measures a cross-section's resistance to bending. Larger I for the same material means less deflection under load.
Why does shape matter more than area?
Material farther from the neutral axis contributes more to stiffness. A hollow section can outperform a solid bar with the same area.
Which axis should I use?
Use the axis about which bending occurs. For symmetric sections, Ixx and Iyy differ — pick the one aligned with your load direction.
What is the difference between section modulus and moment of inertia?
Moment of inertia (I) measures a section's resistance to bending and has units of length⁴. Section modulus S = I/c divides I by the distance c from the neutral axis to the extreme fibre, giving a length³ value that links directly to bending stress via σ = M/S. The calculator also reports the radius of gyration k = √(I/A), used in column-buckling checks.
Put section properties to work in stress analysis