Area by Coordinates (Shoelace)
Finds the area and perimeter of a parcel, building pad or other straight-sided figure of 3 to 6 corners from the coordinates of its corners. Surveyors use this coordinate method to compute the area of a closed traverse for a deed or plat [1]. It also suits takeoffs of paving, fill or planting areas from a site plan.
How it is calculated
The shoelace formula (the coordinate method) sums the cross products of each pair of neighboring corners, going around the figure once:
2A = Σ (Xi·Yi+1 − Xi+1·Yi)
where Xi and Yi are the easting and northing of corner i, and the last corner pairs with the first to close the figure. A is half the absolute value of the sum. The sum is positive when the corners run counter-clockwise and negative when they run clockwise, which is how the page reports the vertex order. The perimeter adds the straight distances between neighboring corners, including the closing side back to corner 1. The area is also given in hectares (10,000 m²) and international acres (4,046.86 m²).
Limits and common mistakes
- Enter the corners in order around the boundary, either way round. Out of order, the sides cross and the area is wrong, with no warning.
- Don’t repeat corner 1 at the end; the page closes the figure itself.
- Curved boundaries are not handled. Add or subtract each curve’s segment area separately.
- Large coordinates such as state plane values work, but subtracting a nearby origin first makes the numbers easier to check.
- The area is on the coordinate grid. On a projected grid, divide by the square of the combined scale factor for the ground area.
- The inputs are in meters. For feet, the area comes out in square feet, and the hectare and acre rows no longer apply.
Worked example
A five-sided lot has been surveyed in a local grid with corner 1 at the origin. The owner needs its area for a property listing.
- Choose Number of Vertices 5.
- Enter the X / Easting and Y / Northing of each vertex in turn: Vertex 1 (0, 0), Vertex 2 (62.5, 0), Vertex 3 (70, 38.2), Vertex 4 (31.4, 55.6) and Vertex 5 (−4.8, 30).
- Press Solve. The Solution shows:
Area: 3,144.45 m²
Area: 0.314445 ha
Area: 0.777011 acre
Perimeter: 218.489 m
Vertex order: counter-clockwise
Check by hand with a box. The corners fit in a rectangle 74.8 m wide (−4.8 to 70) and 55.6 m tall, which is 4,158.88 m². The four right triangles left in its corners are 143.25, 335.82, 463.36 and 72.00 m², together 1,014.43 m². The lot is 4,158.88 − 1,014.43 = 3,144.45 m², as the page gives.
