Land Parcel Area from Coordinates (Shoelace Formula)
Land Parcel Area from Coordinates (Shoelace Formula)
When a parcel boundary is defined by surveyed corner coordinates rather than simple rectangular dimensions, its area is computed with the shoelace (surveyor’s) formula — a direct algebraic sum over the coordinate pairs that works for any simple polygon, regular or irregular.This is exactly how legal descriptions and plat area figures are actually computed from a closed traverse, so it is a routine step in subdivision platting, easement area calculations, and boundary surveys.
For a closed polygon the area is A = ½|Σ(x_i·y_{i+1} − x_{i+1}·y_i)|, summed around the boundary and taken as an absolute value. where (x_1,y_1) through (x_4,y_4) are the surveyed corner coordinates of the parcel, taken in order around its boundary.
Inputs
x_1 := 0 m=0 mm
y_1 := 0 m=0 mm
x_2 := 40 m=40 m
y_2 := 0 m=0 mm
x_3 := 45 m=45 m
y_3 := 30 m=30 m
x_4 := 5 m=5 m
y_4 := 25 m=25 m
The shoelace sum cross-multiplies consecutive coordinate pairs around the closed boundary; taking half the absolute value of the total gives the enclosed area regardless of parcel shape.
A_parcel := 0.5·abs((x_1·y_2-x_2·y_1)+(x_2·y_3-x_3·y_2)+(x_3·y_4-x_4·y_3)+(x_4·y_1-x_1·y_4))=1087.5 m^2
Results
A result of about 1088 m² is a plausible small urban lot area for a four-sided parcel of this size. The method generalizes directly to polygons with more corners by extending the same cross-multiply-and-sum pattern, which is why it is the standard hand and software method for computing area from a closed traverse.
Assumptions & limitations
- Coordinates must be listed in consistent order (all clockwise or all counterclockwise) around the boundary.
- Simple (non-self-intersecting) polygon assumed.
- Flat-plane (local coordinate) geometry — curvature of the earth not considered.