Moment of Inertia Calculator
Calculate the area moment of inertia — the second moment of area — for the common cross-sections used in beam bending: rectangle, solid circle, hollow tube and solid shaft. This geometric property, in units of length to the fourth power, tells you how well a shape resists bending or twisting; material placed far from the neutral axis counts for much more, which is why I-beams and tubes are efficient. The tool shows the formula and the numbers going into it. This is the area moment used in bending, not the mass moment used in rotational dynamics.
How to use this tool
- Choose the cross-section shape.
- Enter its dimensions in millimetres — the fields you need depend on the shape.
- Read the moment of inertia in mm⁴, cm⁴ and m⁴, with the formula and numbers shown.
- For bending shapes, the section modulus S = I/c is also given.
- Use I in beam-deflection and bending-stress calculations; use the polar J for torsion of a shaft.
The formula
The area moment of inertia (second moment of area) measures how a cross-section resists bending. It depends only on the shape and how its material is distributed about the neutral axis — material far from the axis contributes far more, as the fourth-power terms show.
rectangle: I = b·h³ / 12
solid circle: I = π·d⁴ / 64
tube: I = π·(D⁴ − d⁴) / 64
solid shaft (polar): J = π·d⁴ / 32- I
- Area moment of inertia, length⁴
- b, h
- Rectangle width and height
- d, D
- Inner and outer diameters
This is the geometric (area) moment of inertia used in beam bending, not the mass moment of inertia used in rotational dynamics — they share a name but measure different things.
Worked examples
Rectangular beam
- Given
- b = 100 mm, h = 200 mm
- Result
- I ≈ 6.67×10⁷ mm⁴
I = b·h³/12 = 100·200³/12 ≈ 66.7 million mm⁴. The cube on height is why deep beams are so much stiffer.
Round bar
- Given
- d = 50 mm
- Result
- I ≈ 3.07×10⁵ mm⁴
I = π·d⁴/64 ≈ 306,800 mm⁴ about any diameter through the centre.
Tube vs solid
- Given
- D = 50, d = 40 mm
- Result
- I ≈ 1.81×10⁵ mm⁴
A tube keeps most of a solid bar’s stiffness at a fraction of the weight, since the removed core was near the axis.
Frequently asked questions
The area moment of inertia predicts how much a beam or shaft bends or twists under load. It appears directly in the deflection and bending-stress formulas: a larger moment of inertia means a stiffer, stronger section for the same material.
Because material far from the neutral axis contributes far more — the formulas cube or raise dimensions to the fourth power. That is why an I-beam puts most of its material in the flanges, top and bottom, and why a tube is nearly as stiff as a solid bar of the same diameter but much lighter.
They share a name but differ. The area moment of inertia is geometric, in length⁴, and governs bending of a cross-section. The mass moment of inertia is about rotational dynamics — how mass resists angular acceleration — in kg·m². This tool computes the area moment.
It is the second moment of area about the axis perpendicular to the cross-section, governing torsion (twisting). For a solid shaft it is J = π·d⁴/32, exactly twice the diametral I. The tool reports J when you choose the shaft option.
Section modulus S = I/c, where c is the distance from the neutral axis to the outermost fibre. It bundles the geometry needed for bending stress: maximum bending stress equals bending moment divided by S. A larger S means lower stress for the same load.
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