Calculator · Engineering Mechanics
Area Moment of Inertia Calculator
Second moments of AREA about the shape's own centroidal axes, in m⁴. These are geometric properties — never a mass moment of inertia in kg·m².
Shape
Centroidal properties
| Area A | 200 cm² |
|---|---|
| Ix (about the centroidal x-axis) | 1667 cm⁴ |
| Iy (about the centroidal y-axis) | 6667 cm⁴ |
| Ixy (product of inertia) | 0 m⁴ |
| Local centroid | (0, 0) m |
Sign convention: Ixy = ∫xy dA. Ix and Iy are always positive for a solid shape; Ixy may be positive, negative or zero, and is zero when either reference axis x or y is an axis of symmetry of the area about the chosen origin.
Parallel-axis translation
| Ix = Ix,c + A·dy² | 1667 cm⁴ |
|---|---|
| Iy = Iy,c + A·dx² | 6667 cm⁴ |
| Ixy = Ixy,c + A·dx·dy | 0 m⁴ |
Principal area properties
| I₁ | 6667 cm⁴ |
|---|---|
| I₂ | 1667 cm⁴ |
| I_avg = (Ix + Iy)/2 | 4167 cm⁴ |
| Principal-axis angle θp | 90° |
Diagnostics
Technical reference
Ix = ∫y² dA
Iy = ∫x² dA
Ixy = ∫xy dA
- Ix, Iy
- second moment of area about the centroidal x- or y-axis[m⁴]
- Ixy
- product of inertia about those same axes[m⁴]
- A
- cross-sectional area[m²]
Ix and Iy — also called the second moment of area — describe how a cross-section's area is distributed for bending and are non-negative; they are positive for a non-degenerate finite-area section. Ixy can be positive, negative or zero, and vanishes when either reference axis x or y is an axis of symmetry of the area about the chosen origin.
For the rectangle this tool implements, Ix,c = b·h³/12 and Iy,c = h·b³/12 about the centroid; the circle and semicircle kernels use their own closed forms rather than a general handbook table.
This is a geometric property in m⁴, never to be confused with the mass moment of inertia in kg·m² used for rotational dynamics.
Assumptions
- · Second moments of area [m⁴]; a mass moment of inertia [kg·m²] is a different quantity and lives in its own calculator.
- · Centroidal values are about the shape's own centroid, using the documented local frame of each kernel.
- · The translation applies the parallel-axis theorem about axes parallel to the centroidal ones.
- · When Ix = Iy and Ixy = 0 every axis is principal, so no specific angle is reported.
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