Moment of Inertia Calculator

Second moment of area, section modulus and radius of gyration for rectangles, solid round bars, pipes/tubes and I-beams.

Measured in the direction of bending (perpendicular to the x-x axis).
Moment of inertia Ix (strong / x-x axis)
6,666.67 cm⁴
Ix in inches
160.167 in⁴
Elastic section modulus Sx = Ix / c
666.67 cm³
40.682 in³
Moment of inertia Iy (weak / y-y axis)
1,666.67 cm⁴
40.042 in⁴
Section modulus Sy
333.33 cm³
Cross-sectional area
200
Radius of gyration rx = √(Ix/A)
57.74

Estimate generated with SortedSums: https://sortedsums.com/moment-of-inertia-calculator/. For informational purposes only; verify before relying on it.

Estimate only. This tool is for informational and educational purposes. Results depend on your inputs and simplifying assumptions, and are not a substitute for professional engineering, design or financial advice. Always verify with a qualified professional and applicable codes before purchasing, building or making decisions.

How it’s calculated

The second moment of area (area moment of inertia) measures how a cross-section resists bending. It is taken about the section’s centroidal axes. The elastic section modulus S = I / c, where c is the distance from the neutral axis to the extreme fiber, gives bending stress σ = M / S.

Rectangle: Ix = b·h³/12 Iy = h·b³/12 Solid circle: I = π·D⁴/64 Pipe / tube: I = π·(D⁴ − d⁴)/64 I-beam: Ix = [b·h³ − (b − tw)·(h − 2tf)³]/12 Iy = [2·tf·b³ + (h − 2tf)·tw³]/12

The I-beam formula treats the section as a rectangle minus the two empty rectangles beside the web, and ignores root fillets and tapered flanges, so it slightly underestimates I for rolled sections. For design, use the tabulated properties of the actual rolled shape.

Example: a 100 mm × 200 mm rectangle bending about its strong axis has Ix = 100 × 200³ ÷ 12 = 6.667 × 10⁷ mm⁴ = 6,667 cm⁴ and Sx = 6.667 × 10⁷ ÷ 100 = 666.7 cm³.

Frequently asked questions

Is this the same as mass moment of inertia?

No. This is the area moment of inertia (units of length⁴) used for beam bending and buckling. Mass moment of inertia (kg·m²) describes resistance to rotational acceleration.

Why does depth matter so much?

For a rectangle I grows with the cube of the depth, so doubling the depth makes the beam 8 times stiffer. That is why joists are installed on edge.

What is the difference between I and S?

I controls deflection (stiffness); S = I/c controls the maximum bending stress (strength).

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Sources

Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.

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