Polygon Area (Shoelace Formula)
Enter the vertices of a simple polygon (one x, y pair per line, in order around the perimeter) and this tool computes the area via the shoelace (surveyor's) formula, the perimeter, the centroid, the winding orientation (clockwise vs counter-clockwise), and whether the polygon is convex or concave. Coordinates are plain numbers — integers or decimals. Runs locally in your browser.
Vertices
Result
The shoelace formula gives the signed area of a polygon from its vertices (xᵢ, yᵢ): A = ½ |Σᵢ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, with indices wrapping. The sign is positive for counter-clockwise winding and negative for clockwise, so the orientation follows the sign of the signed area (zero means the vertices are collinear / degenerate). The centroid is the area-weighted center Cx = (1/6A) Σ (xᵢ+xᵢ₊₁)(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) and likewise for Cy. Convexity is checked by testing that every successive cross product turns the same way. The polygon is assumed simple (non-self-intersecting); a self-intersecting polygon gives a signed "algebraic" area that does not match its visual area. Pairs with the Triangle Solver and Coordinate Converter. Everything runs locally — nothing leaves your browser.