Area Moment of Inertia Calculator
Calculate centroidal second moments of area about the horizontal and vertical axes for six common cross-section shapes.
Cross-section inputs
Iₓ = b·h³/12; Iᵧ = h·b³/12
Choose a shape and enter every required dimension to calculate Iₓ and Iᵧ.
Centroidal axis comparison
The bars compare the two area moments in the same selected fourth-power unit. A longer bar indicates that area is distributed farther from that axis.
Iᵧ is 2.25 times Iₓ for the startup rectangle.
Formula and result details
| Axis | Centroidal formula | Result |
|---|---|---|
| Horizontal x-axis (Iₓ) | b·h³/12 | 512 cm⁴ |
| Vertical y-axis (Iᵧ) | h·b³/12 | 1,152 cm⁴ |
These are centroidal area moments, not mass moments of inertia. For an axis shifted away from the centroid, apply the parallel-axis theorem separately.
How to use the area moment of inertia calculator
What this calculator does
This calculator estimates the centroidal second moment of area, commonly written as Iₓ and Iᵧ, for a triangle, rectangle, circle, semicircle, ellipse, or regular hexagon. It measures how a two-dimensional area is distributed around the horizontal and vertical axes that pass through its centroid. The result is a geometric property in a fourth-power length unit such as cm⁴ or in⁴. It is not the mass moment of inertia used for rotational dynamics, and it does not by itself determine stress, deflection, material strength, or whether a real member is safe. The underlying definitions and common-shape derivations are explained in the Engineering Statics treatment of moments of inertia for common shapes.
When to use it
Use the calculator to compare candidate cross-section orientations, verify hand calculations for a standard shape, prepare inputs for a beam-bending or deflection model, or check how strongly a dimension change affects an axis. It is particularly useful when a rectangular or elliptical section may be rotated, because swapping horizontal and vertical dimensions also swaps which centroidal moment is larger.
How to calculate
- The calculator opens with a complete demonstration: a 12 cm by 8 cm rectangle. Its finite results and a validated example XLSX workbook are immediately available.
- Select a Shape. The relevant dimension fields appear and unrelated fields are disabled.
- Choose the Length unit. If dimensions are already entered, the calculator converts them so the physical size is preserved.
- Replace the demonstration dimensions with your own positive values. Use a decimal point; grouped values such as 1,250.5 are accepted, while ambiguous decimal-comma input such as 1,5 is rejected.
- Read Moment about the horizontal centroidal axis, Iₓ and Moment about the vertical centroidal axis, Iᵧ. The comparison chart and formula table use the same canonical values.
- Select Download Excel to create a current-state OOXML workbook. Reset clears the demonstration and all dimensions; export is then disabled until a complete valid shape is entered again.
Input guide
Shape and Length unit
Shape is required and selects one of six formula sets. For example, choose Rectangle for a solid 12 by 8 section. Length unit is required and accepts mm, cm, m, in, or ft. Changing it converts existing dimensions and changes the displayed fourth-power unit. A common mistake is mixing dimension units; convert all dimensions to the same selected unit before interpreting the output.
Width (b) and Height (h)
These required positive decimals appear for rectangles and triangles. Width is horizontal and height is vertical; 12 cm and 8 cm are the startup values. Increasing height strongly raises Iₓ because height is cubed, while increasing width strongly raises Iᵧ. Do not enter full dimensions in one unit while the selector shows another, and do not use zero or negative lengths.
Top vertex displacement (a)
This required triangle-only decimal locates the top vertex horizontally from the left base vertex. It must be between 0 and Width (b), inclusive; a value of b/2 gives a symmetric isosceles triangle. It changes Iᵧ but not Iₓ. The frequent mistake is treating a as the triangle height or as an offset from the center rather than from the left end of the base.
Radius (r)
This required positive decimal appears for circles and semicircles; 4 cm is a realistic example. Both results scale with r⁴, so doubling the radius multiplies each applicable moment by 16. For a semicircle, Iₓ is taken about its centroidal axis parallel to the flat edge. Enter the radius, not the diameter.
Horizontal and Vertical semi-axes
Horizontal semi-axis (rₓ) and Vertical semi-axis (rᵧ) are required positive ellipse dimensions. Values of 6 cm and 3 cm describe a 12 cm by 6 cm ellipse. Each is half the full dimension. Increasing rᵧ has a cubic effect on Iₓ, while increasing rₓ has a cubic effect on Iᵧ.
Side length (s)
This required positive decimal is used only for a regular hexagon, where all six sides are equal. A 5 cm side is a typical example. Both centroidal moments are equal by symmetry and scale with s⁴. Do not substitute the distance across flats or across opposite vertices for the side length.
Output guide
Moment about the horizontal centroidal axis, Iₓ quantifies area spread in the vertical direction around the horizontal axis. Moment about the vertical centroidal axis, Iᵧ quantifies area spread in the horizontal direction around the vertical axis. Both are exact identities for the selected idealized shape and entered dimensions, subject only to displayed rounding. Zero is not a valid result for a positive solid shape; a very large result usually reflects a large dimension raised to the fourth power. The header pills Shape, Axes, and Output restate the active formula set, the centroidal-axis assumption, and the fourth-power unit. The Current formula set, formula summary, axis-comparison bars, chart summary, and the table's Axis, Centroidal formula, and Result columns all identify the same two computed moments. A larger bar means the corresponding moment is larger; it does not directly state allowable load or stiffness without the rest of an engineering model.
Worked example
For the startup rectangle, Width (b) is 12 cm and Height (h) is 8 cm. The horizontal-axis calculation is Iₓ = b·h³/12 = 12·8³/12 = 512 cm⁴. The vertical-axis calculation is Iᵧ = h·b³/12 = 8·12³/12 = 1,152 cm⁴. Therefore Iᵧ is 2.25 times Iₓ, which is why the Iᵧ bar is longer. The values match the first-open controls, result cards, formula table, chart labels, and exported workbook.
Learn more
The formulas here assume axes through the centroid. To evaluate a parallel axis elsewhere, use I = Icentroid + A·d² as described in the parallel-axis theorem. For shapes that are not covered by a standard formula, the second moment is obtained by area integration; OpenStax's calculus discussion of moments and double integrals provides the broader mathematical context.
Interpreting the result in practice
Area moment of inertia is highly sensitive to section depth because one dimension is raised to the third power in rectangular formulas and all characteristic lengths effectively contribute to a fourth-power scale. That is why a small change in orientation can produce a large change between Iₓ and Iᵧ. In beam analysis, the applicable value is the one about the axis of bending. Keep the axis convention consistent with the drawing, and verify whether a source uses full width and height, radii, semi-axes, or another geometric measure.
This calculator covers single, solid, idealized shapes about their centroidal axes. Composite sections, holes, rotated non-principal axes, and offset axes require additional centroid, subtraction, rotation, or parallel-axis calculations.