What does the calculator compute?
The calculator determines the principal moments of inertia I1 and I2, and the corresponding principal axis angle φ, for any polygonal cross-section contour. The centroid and Ix, Iy and Ixy are found internally with the same polygon engine used by the second-moment and centroid calculators.
From Ix, Iy, Ixy to the principal values
The principal moments of inertia follow from:
I1,2 = (Ix+Iy)/2 ± √(((Ix−Iy)/2)² + Ixy²)The corresponding principal axis angle follows from tan(2φ) = −2Ixy/(Ix−Iy). Since this equation has two solutions 90° apart, the calculator additionally checks which of the two axes actually belongs to I1, rather than assuming a particular sign convention.
Control value I1 + I2
The sum of the principal moments of inertia is independent of the axis rotation and must always equal Ix + Iy (the polar moment of inertia about the centroid). This invariant is shown on the results panel and serves as a plausibility check.
Symmetric cross-sections
If the cross-section has an axis of symmetry that coincides with the x- or y-axis, Ixy = 0 and the xS/yS axes shown are already the principal axes. For an unsymmetric profile, the principal axes are instead rotated by the angle φ relative to xS/yS.
Example: 100 × 50 mm rectangle rotated by 25°
An axis-aligned 100 × 50 mm rectangle has Ix = 1,041,666.667 mm⁴, Iy = 4,166,666.667 mm⁴ and Ixy = 0 -- the principal axes already coincide with x/y (I1 = Iy, φ = 90°, since I1 always marks the axis with the larger value). Rotating the same contour by 25° about its centroid leaves I1 = 4,166,666.667 mm⁴ and I2 = 1,041,666.667 mm⁴ unchanged, while the principal axis angle shifts by the same amount to φ = 115° -- the principal axes are a property of the shape and rotate with it.
Scope and limitations
The calculator supports one closed, non-self-intersecting outer contour and one polygonal inner contour lying completely inside it. Disconnected regions, arcs and DXF are currently excluded.