Polygon Shoelace Area

Polygon area — shoelace formula

Give the shoelace formula (sometimes called Gauss's area formula) the coordinates of a polygon's vertices and it returns the area. Any simple polygon works, whether convex or concave, regular or irregular: A = ½·|Σᵢ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, with indices taken modulo n. Drop the absolute value and the sign tells you something useful. It comes out positive when the vertices run counterclockwise and negative when they run clockwise, so you can read off the orientation directly. Take vertices (0,0), (4,0), (4,3), (2,5), (0,3): A = ½·|(0·0 − 4·0) + (4·3 − 4·0) + (4·5 − 2·3) + (2·3 − 0·5) + (0·0 − 0·3)| = ½·|0 + 12 + 14 + 6 + 0| = 16.

Applications

It's a workhorse in GIS (geographic information systems), where it measures georeferenced plots, drainage basins and protected zones. CAD software leans on it for polygon properties, and computational geometry uses it inside convex-hull and triangulation routines. In Brazil, INCRA runs shoelace-style calculations on the SIGEF system to check rural plot areas declared in georeferenced certifications. Architects reach for it too, pulling floor-plan areas straight from CAD vertex lists.

FAQ

Does the order of vertices matter? It does. List them sequentially around the perimeter, all clockwise or all counterclockwise. Scramble that order and you'll get a different area, almost always a wrong one.

Does it work for concave polygons? Yes, provided the polygon is simple, meaning its edges never cross. When edges do intersect, the result is an algebraic sum of signed areas rather than the geometric area you'd expect.

Why "shoelace"? Lay the coordinates out in two columns and connect the cross-products, and the crisscross pattern looks like the lacing on a shoe. That's where the name comes from.

How does the sign indicate orientation? A positive signed sum means the vertices run counterclockwise; a negative one means clockwise. GIS tools use this to keep a consistent winding order across shapefiles.