Polygon Shoelace Area Calculator

Method, example, and limits

Formula: Shoelace: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. Perimeter is Σ√((xᵢ₊₁−xᵢ)² + (yᵢ₊₁−yᵢ)²), including the closing edge.

Inputs and units: Each line contains one vertex as x,y. Coordinates must use one consistent linear unit; area is in the square of that unit and perimeter in the original unit.

Worked example: The rectangle (0,0), (4,0), (4,3), (0,3) has area 12 square units and perimeter 14 units.

Assumptions: Vertices are listed once in order around a planar, closed, simple polygon. Enter 3–200 finite coordinate pairs; the closing edge from the last point to the first is implicit.

Limits: Self-intersecting polygons, repeated adjacent vertices, zero-area outlines, geographic latitude/longitude coordinates, and coordinates beyond ±1,000,000,000 are rejected. Coordinates are not projected or geodesic.