Polygon Area Shoelace Calculator

Polygon area from vertices — shoelace formula

Give the shoelace formula (you may also see it called Gauss's area formula) the coordinates of a polygon's vertices in order and it hands back the area: A = ½·|Σᵢ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, with indices taken modulo n. Convex or concave, it doesn't matter, as long as the edges never cross each other. Drop the absolute value and the signed sum tells you the orientation too: a positive number means you entered the vertices counterclockwise, a negative one means clockwise. Take the rectangle (0,0), (4,0), (4,3), (0,3): A = ½·|(0·0−4·0) + (4·3−4·0) + (4·3−0·3) + (0·0−0·3)| = ½·|0 + 12 + 12 + 0| = 12.

Applications

It's the workhorse behind area figures in GIS tools like QGIS and ArcGIS, whether you're measuring georeferenced plots, drainage basins, conservation zones or built-up footprints. Brazil's INCRA SIGEF system leans on the same shoelace-style math to validate the area of rural plots during georeferenced certification. You'll also find it inside CAD packages (AutoCAD, BricsCAD), pulling polygon areas out of vertex lists; in game development, where collision and visibility polygons need it; and across computational geometry, in hull and triangulation routines.

FAQ

Does it work for concave polygons? It does, for any simple polygon (edges that don't intersect themselves), concave ones included. Once edges start crossing, what you get back is an algebraic sum of signed areas rather than the geometric area.

Does the order of vertices matter? It does. List them as you'd walk the perimeter, all clockwise or all counterclockwise. Scramble that order and the number you get is meaningless.

What does the sign of the unsigned-out result mean? A positive sum means the vertices run counterclockwise; negative means clockwise. GIS relies on that to keep winding canonical, for instance CCW for outer rings and CW for holes in shapefiles.

Does it close the polygon automatically? Yes. The modulo on the indices wraps the last vertex back to the first, so there's no need to repeat the opening point at the end.