About Area by Coordinates Calculator
The area by coordinates calculator returns the enclosed area of a closed traverse or land parcel directly from the coordinates of its corners. It applies the Shoelace formula (also called the Surveyor’s or Gauss area formula), which sums the cross-products of successive vertex coordinates and halves the absolute value, so the parcel does not need to be broken into triangles or trapezoids by hand.
Enter the vertices in order as (x, y) or (Easting, Northing) pairs in metres. The tool computes the unsigned area, converts it to hectares, acres, and square feet, sums the edge lengths to give the perimeter, locates the area centroid, and reports whether the vertices were listed clockwise or counter-clockwise. Both orderings give the same area because the formula uses the absolute value of the signed sum.
How It Works
- List the polygon vertices in order around the boundary as (Easting, Northing) coordinates; at least three are required and the figure is closed automatically from the last vertex back to the first.
- The calculator evaluates the Shoelace sum A = 0.5 |sum(x_i * y_{i+1} - x_{i+1} * y_i)| over all edges.
- It sums the straight-line lengths of every edge to give the perimeter and evaluates the polygon area-centroid formulas.
- The sign of the un-absolute-valued sum sets the winding direction: positive is counter-clockwise (CCW), negative is clockwise (CW), in a standard East-North plane.
Worked Example
A rectangular parcel has corners at (0, 0), (50, 0), (50, 40), and (0, 40) metres, listed counter-clockwise. The Shoelace sum is 0.5 |(0*0 - 50*0) + (50*40 - 50*0) + (50*40 - 0*40) + (0*0 - 0*40)| = 0.5 |0 + 2000 + 2000 + 0| = 2000 m², which equals 0.2000 hectares and 0.4942 acres. The perimeter is 50 + 40 + 50 + 40 = 180 m, the centroid is at (25, 20), and the order is CCW.
Formulas
- Shoelace (Surveyor’s) area
A = 0.5 * |sum_{i=1}^{n} (x_i * y_{i+1} - x_{i+1} * y_i)|- Signed area and winding direction
A_s = 0.5 * sum (x_i * y_{i+1} - x_{i+1} * y_i); A_s > 0 => CCW, A_s < 0 => CW- Polygon area centroid
Cx = 1/(6 A_s) * sum (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i); Cy = 1/(6 A_s) * sum (y_i + y_{i+1})(x_i y_{i+1} - x_{i+1} y_i)- Perimeter
P = sum sqrt((x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2)
Standards & References
- Shoelace / Surveyor’s (Gauss) area formula
- Coordinate (plane) surveying
- Ghilani & Wolf, Elementary Surveying
Frequently Asked Questions
Does the vertex order (clockwise vs counter-clockwise) matter for the area?
No. The Shoelace formula takes the absolute value of the signed sum, so clockwise and counter-clockwise orderings give the same area. The sign is only used to report the winding direction, which the tool shows as CW or CCW.
How many vertices do I need and do I have to repeat the first point?
You need at least three vertices to enclose an area. You do not repeat the first point; the calculator closes the polygon automatically by connecting the last vertex back to the first.
What units does the area come out in?
Enter coordinates in metres. The area is reported in square metres and also converted to hectares (1 ha = 10,000 m²), acres (1 acre = 4046.8564 m²), and square feet (1 m² = 10.7639 ft²).
Why is the centroid not just the average of the corner coordinates?
The reported centroid is the area centroid of the polygon, which weights the geometry of the whole figure, not the simple mean of the vertices. For a non-symmetric or L-shaped parcel the two differ, so the area-weighted centroid is the correct location of the parcel’s center of area.