Second moment of area for arbitrary cross-sections

What does the calculator compute?

The calculator determines geometric section properties for any straight-edged 2D outer contour. Unlike the calculator for rectangles, circles or tubes, the shape is described by vertices. Enter them in boundary order; the final vertex is connected logically to the first.

What is the second moment of area?

The second moment of area describes how area is distributed relative to an axis. In beam bending, EI is the flexural rigidity: E is a material property and I is purely geometric. A large I therefore indicates favourable area distribution about the chosen bending axis; it does not by itself prove strength.

A flat rectangle illustrates the axis dependence: its values about the strong and weak centroidal axes differ greatly. I is therefore meaningful only together with its reference axis.

Polygon integrals and sign convention

For each edge the engine forms ci = xi·yi+1 − xi+1·yi. The sums give area, centroid and origin-based integrals. A clockwise contour is reversed internally. The product moment convention used here is Ixy = ∫A x·y dA, which may be positive, zero or negative.

Centroidal axes and parallel-axis theorem

After finding the centroid, the moments are transformed to parallel axes through S. This makes the primary result independent of the chosen coordinate origin.

Example: 100 × 50 mm rectangle

The initial contour (0,0), (100,0), (100,50), (0,50) gives A = 5,000 mm² and S = (50 mm, 25 mm). Analytically, Ix = b·h³/12 = 1,041,666.667 mm⁴ and Iy = h·b³/12 = 4,166,666.667 mm⁴. Both symmetry axes make Ixy = 0. These values are also used as an automated reference test.

Scope and limitations

The calculator supports one closed, non-self-intersecting outer contour made of straight edges. Holes, disconnected regions, arcs, DXF, plastic properties and torsion are currently excluded. Geometric properties alone do not verify strength, stability or serviceability.