Land Area Calculator
Area From an Irregular Boundary, Not Just a Rectangle
Real parcels of land are rarely clean rectangles — property lines follow old field boundaries, rivers, and road curves. Once you have the boundary as a list of vertex coordinates, the shoelace formula computes the exact enclosed area of any simple polygon, no matter how irregular, without needing to break it into triangles by hand.
The Formula
The formula sums a cross-product term across each consecutive pair of vertices (wrapping back to the first point at the end), then halves the absolute value — a direct application of the Gauss shoelace method, so named because tracing the cross-multiplication pattern across a list of coordinates resembles lacing a shoe.
Worked Examples
| Vertices | Shape | Area |
|---|---|---|
| (0,0); (4,0); (4,3); (0,3) | Rectangle | 12 sq units |
| (0,0); (6,0); (3,4) | Triangle | 12 sq units |
Where This Calculation Matters
- Property and parcel surveys — computing the area enclosed by a set of surveyed boundary points.
- Agricultural field measurement — calculating the area of an irregularly shaped field from GPS-walked or drone-mapped boundary points.
- GIS polygon area calculations — the shoelace formula is the standard method used under the hood by most mapping and CAD software for computing polygon area.
How to Use This Calculator
- Enter the boundary vertex coordinates as x,y pairs, one per line or separated by semicolons — e.g.
0,0; 4,0; 4,3; 0,3. - List at least 3 vertices in order around the polygon's perimeter.
- Select Calculate to get the enclosed area.
Related Calculations
Once you have an area, work out crowding with the Population Density Calculator, or scale a plotted site plan with the Map Scale Calculator.