How to use the area moment of inertia calculator
- Select the cross section shape from the dropdown menu.
- Enter the required dimensions such as width, height, diameter or wall thickness.
- The area moment of inertia (Ix, Iy) and section modulus update automatically.
- Use consistent units for all inputs; the output units are your input units raised to the fourth power.
What is the area moment of inertia?
The area moment of inertia, also called the second moment of area, measures how a cross section resists bending. It depends only on geometry, not on material. A larger value means the beam deflects less under load. Engineers denote it Ix for bending about the horizontal axis and Iy for the vertical axis. The elastic section modulus S equals I divided by the distance from the neutral axis to the outermost fibre, giving the ratio of bending moment to maximum stress.
Formulas used in this calculator
Each shape uses a closed-form equation derived from integration over the area. For a solid rectangle with width b and height h:
Ix = b * h^3 / 12
Iy = h * b^3 / 12
For example, a 100 mm wide and 200 mm tall rectangle gives Ix = 100 * 200^3 / 12 = 66,666,666.667 mm^4. The section modulus is Sx = Ix / (h/2) = 666,666.667 mm^3.
A solid circle of diameter d has Ix = Iy = pi * d^4 / 64. A hollow circle subtracts the inner void: Ix = pi * (d^4 - di^4) / 64. A hollow rectangle works the same way: Ix = (b * h^3 - bi * hi^3) / 12.
An I-beam is computed by taking the full bounding rectangle and subtracting the two side gaps beside the web. With overall width b, height h, web thickness tw and flange thickness tf, the gap width on each side is (b - tw) / 2 and the gap height is (h - 2*tf). The formula becomes Ix = (b * h^3 - (b - tw) * (h - 2*tf)^3) / 12.
Reference table of common sections
The table below shows pre-computed values for typical dimensions in millimetres. Use it to verify your own calculations or to compare shapes quickly.
| Shape | Dimensions (mm) | Ix (mm^4) | Sx (mm^3) |
|---|---|---|---|
| Rectangle | b=50, h=100 | 4,166,667 | 83,333 |
| Circle | d=50 | 306,796 | 12,272 |
| Hollow circle | d=100, di=80 | 2,898,119 | 57,962 |
| I-beam | b=100, h=200, tw=10, tf=15 | 29,819,167 | 298,192 |
Practical applications
Structural engineers use the second moment of area to size beams in buildings, bridges and frames. When you know the applied bending moment M and the allowable stress, you can find the minimum required section modulus S = M / stress, then pick a shape that meets it. Mechanical designers apply the same logic to shafts, axles and machine frames. Because the area moment of inertia grows with the cube or fourth power of depth, adding height to a beam is far more effective than adding width. That is why I-beams concentrate material in the flanges away from the neutral axis.
Common mistakes to avoid
The most frequent error is mixing units. If you enter width in centimetres and height in millimetres, the result is meaningless. Always convert every dimension to the same unit before calculating. Another mistake is confusing the area moment of inertia with the mass moment of inertia used in dynamics; they share a name but have different units and purposes. Finally, remember that these formulas assume the centroidal axes. If your reference axis is offset, you must apply the parallel axis theorem: I = I_centroid + A * d^2, where A is the area and d is the offset distance.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Solid rectangle 100 x 200 | Cross section shape: rectangle; Width b: 100; Height h: 200 | Ix: 66,666,666.667; Iy: 16,666,666.667; Sx: 666,666.666667; Sy: 333,333.333333 |
| Solid circle d=50 | Cross section shape: circle; Diameter d: 50 | Ix = Iy: 306,796.157577; Sx = Sy: 12,271.846303 |
| Hollow rectangle 100x200, inner 80x160 | Cross section shape: hollow-rect; Width b: 100; Height h: 200; Inner width bi: 80; Inner height hi: 160 | Ix: 39,360,000; Iy: 9,840,000; Sx: 393,600; Sy: 196,800 |
| Hollow circle D=100, di=80 | Cross section shape: hollow-circle; Diameter d: 100; Inner diameter di: 80 | Ix = Iy: 2,898,119.223; Sx = Sy: 57,962.384459 |
| I-beam b=100, h=200, tw=10, tf=15 | Cross section shape: i-beam; Width b: 100; Height h: 200; Web thickness tw: 10; Flange thickness tf: 15 | Ix: 29,819,166.667; Iy: 2,514,166.667; Sx: 298,191.666667; Sy: 50,283.333333 |
FAQ
What is the difference between Ix and Iy?
Ix is the second moment of area about the horizontal centroidal axis, governing resistance to vertical bending. Iy is about the vertical axis, governing lateral bending. For symmetric shapes like circles they are equal.
What units does this area moment of inertia calculator use?
The calculator is unit-agnostic. If you enter dimensions in millimetres, the result is in mm^4. Enter inches and you get in^4. Just keep all inputs in the same unit.
How is section modulus related to moment of inertia?
Elastic section modulus S equals I divided by c, where c is the distance from the neutral axis to the extreme fibre. For a rectangle of height h, c = h/2, so Sx = Ix / (h/2).
Does this tool account for material properties?
No. The area moment of inertia is purely geometric. To find actual deflection or stress you also need the material's Young's modulus and the applied loads.
Why is an I-beam more efficient than a solid rectangle?
An I-beam places most of its material far from the neutral axis, which increases Ix dramatically while using less total area. This gives a higher section modulus per unit weight.
Can I use this for asymmetric sections?
This calculator covers doubly symmetric shapes only. For channels, angles or T-sections you need to locate the centroid first, then apply the parallel axis theorem to each part.