Area moment of inertia calculator
This area moment of inertia calculator covers the common cross-sections: enter the dimensions and get Iy, Iz, the section moduli and the area, optionally with the parallel axis term for a shifted axis.
Example: Cross-section: Rectangle · Unit of the dimensions: mm · Width b 60 · Height h 120 → Result: I_y = 8,640,000 mm⁴ · W_y = 144,000 mm³ · A = 7,200 mm². Source: Wikipedia – Flächenträgheitsmoment (formulas for rectangle, circle, tube, triangle, I-section). Updated: .
How it is calculated
The area moment of inertia of a rectangle is Iy = b·h³ / 12. A 60 × 120 mm beam has Iy = 8,640,000 mm⁴ (864 cm⁴) and a section modulus Wy = 144,000 mm³.
What it is used for
The area moment of inertia (second moment of area) measures how stiff a cross-section is against bending. With the modulus of elasticity it gives the deflection, and with the section modulus W = I / e the bending stress σ = M / W. Depth enters to the third power: a beam twice as deep is eight times as stiff.
Formulas for the sections
- Rectangle: Iy = b·h³/12, Wy = b·h²/6
- Circle: I = π·d⁴/64, W = π·d³/32
- Circular ring: I = π·(D⁴ − d⁴)/64
- Triangle: Iy = b·h³/36
- Box and I-beam: difference or sum of rectangles
If the reference axis lies parallel to the centroidal axis at distance a, the parallel axis theorem applies: I′ = I + A·a².
Frequently asked questions
How do you calculate the area moment of inertia of a rectangle?
With Iy = b·h³/12 for bending about the horizontal axis, where h is the depth in the load direction. About the vertical axis, Iz = h·b³/12.
What is the difference from the mass moment of inertia?
The area moment of inertia (unit mm⁴, m⁴) describes the bending stiffness of a section. The mass moment of inertia (kg·m²) describes resistance to angular acceleration.
How does the depth affect it?
To the third power: doubling the depth makes the second moment of area eight times larger and the section modulus four times larger.
Sources and legal basis
- Wikipedia – Flächenträgheitsmoment (formulas for rectangle, circle, tube, triangle, I-section)
- Wikipedia – Satz von Steiner (Verschiebesatz)
As of: · Editorial policy
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