Polygon Area Calculator

Area from a list of corners

Given the corners of a parcel in order around its boundary, the area can be computed without drawing it. On a flat grid, the shoelace formula adds up cross-products of successive vertices. On the globe, the same idea works with latitude and longitude: the area of a polygon whose edges are great-circle arcs is R² ÷ 2 times the absolute sum, around the boundary, of the change in longitude multiplied by (2 + sin φ₁ + sin φ₂). This is the formula used by Google Maps and the Turf.js library. This calculator uses the authalic radius, 6,371,007.2 m, which gives a sphere with exactly the surface area of the WGS84 ellipsoid, keeping area errors well under 0.5% for most parcels.

A worked example

Central Park in New York is close to a rectangle. Taking its four corners from a map in order, 40.7681, −73.9819 at Columbus Circle; 40.8006, −73.9581 at Frederick Douglass Circle; 40.7968, −73.9493 at the north-east corner; and 40.7644, −73.9730 at Grand Army Plaza, gives an area of 3.522 km², 352.2 hectares or 870 acres, with a perimeter of 9.96 km. The park’s published area is 843 acres. The 3% difference comes from the corner points: read from a map, they sit on the surrounding streets rather than exactly on the park wall, which shows how sensitive area is to the boundary you trace. Moving every side out by just 10 m on a park this size adds about 10 hectares, 25 acres.

Units and conversions

A hectare is 10,000 m², a square 100 m on a side. An acre is 4,046.86 m², originally the area a team of oxen could plough in a day, and there are 640 acres in a square mile and 2.471 acres in a hectare. A square kilometre is 100 hectares or 247.1 acres. Property records in the United States use acres, most of the world uses hectares, and the area of lakes and parks is often quoted in square kilometres or square miles.

Sources of error

The vertices must be listed in order around the boundary, either clockwise or anticlockwise; listing them out of order creates a self-crossing shape and a meaningless area. The formula assumes straight edges between vertices, so curved boundaries such as a lakeshore need enough points to follow the curve. Consumer GPS positions are accurate to about 3 to 5 m, so walking the boundary of a 1-hectare field can give an area wrong by a few per cent. A common misconception is that GPS area measurements are good enough for legal purposes; property areas are defined by the deed and a licensed survey, not by a phone.

How to use it

1. Collect the corners from a map, a GPS track or survey notes.
2. Enter one “latitude, longitude” pair per line, in order around the boundary.
3. Do not repeat the first point at the end; the shape is closed automatically.
4. Read the area in square metres, hectares, acres and square miles, and the perimeter.