Area from coordinates calculator: shoelace formula for survey plots

Updated: September 30, 2026 · ISKRIT

The area of a plot from its corner coordinates comes from the shoelace (surveyor's) formula: twice the area equals the sum of Xi · Yi+1 - Xi+1 · Yi taken around the boundary. The calculator below takes grid coordinates in metres or latitude and longitude pasted from a web map, and returns the area in square metres, hectares, acres and square feet together with the perimeter.

Calculator

How it works

Shoelace formula. For every vertex multiply its X by the Y of the next vertex and subtract the X of the next vertex times its own Y. The sum over all vertices is twice the area with a sign that shows the direction of travel, so the absolute value is taken. The order of the two coordinates in a line does not change the area: easting and northing or northing and easting give the same result.

Vertex order. Enter the corners one after another along the boundary, clockwise or anticlockwise. If the order is mixed up the outline crosses itself and the area is wrong. The first corner does not need to be repeated at the end.

Latitude and longitude. Geographic coordinates are converted to metres on the WGS84 ellipsoid at the mean latitude of the plot, where a degree of latitude and a degree of longitude have different lengths. For plots up to tens of kilometres across the difference from a full ellipsoidal computation is below a hundredth of a percent. Lines such as "51.507351, -0.127758" can be pasted as they come from a map.

Legal area. Title documents usually state the area computed from the projected grid coordinates of the boundary survey. Use those coordinates for legal work; points picked on a map are good for estimates.

Worked example

Corners, easting and northing in metres: (512040.00, 6210300.00), (512070.00, 6210300.00), (512078.50, 6210345.20), (512035.20, 6210351.00). Shifted to the first corner they become (0, 0), (30, 0), (38.5, 45.2), (-4.8, 51.0). The shoelace terms are 0, 1356, 2180.46 and 0, the sum is 3536.46, and the area is 3536.46 / 2 = 1768.23 m², which is 0.1768 ha or 0.4369 acres.

Questions

How many points are needed? At least three. The outline closes by itself.

How do I convert to acres? One acre is 4046.856 m² and one hectare is 10 000 m². The calculator shows all units at once.

Why does my map measurement differ from the deed? Points picked on a map are off by metres, while the deed area comes from the boundary survey coordinates.

Cut and fill volumes from survey points inside a boundary are computed by the earthwork volume calculator.