Polygon Area Calculator (Shoelace Formula)
Calculate the area of any polygon from vertex coordinates using the shoelace formula.
Supports triangles, quadrilaterals, and polygons up to 8 vertices.
Polygon Area — Shoelace Formula
The shoelace formula (also called the surveyor’s formula or Gauss’s area formula) computes the signed area of any simple polygon directly from its vertex coordinates. It works for triangles, quadrilaterals, and arbitrary n-sided polygons — convex or concave — as long as the edges do not cross themselves.
Formula
Area = ½ |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|
The sum runs over all vertices, with the last vertex wrapping back to the first. The absolute value handles both clockwise and counter-clockwise vertex orderings.
Why “Shoelace”?
Writing the coordinates in two columns and cross-multiplying diagonally produces a pattern that resembles lacing a shoe — hence the name.
Example
For a triangle with vertices (0,0), (4,0), (0,3):
- Cross-products: 0·0 − 4·0 = 0; 4·3 − 0·0 = 12; 0·0 − 0·3 = 0
- Sum = 12; Area = ½ × |12| = 6 square units
This matches the standard formula ½ × base × height = ½ × 4 × 3 = 6.
Vertex Ordering
Enter vertices in order around the polygon — either clockwise or counter-clockwise. Skipping or reordering vertices breaks the formula because edges become crossed. Many surveying applications use the signed result (without absolute value) to detect orientation: positive means counter-clockwise, negative means clockwise.
Applications
| Field | Use |
|---|---|
| Land surveying | Plot area from GPS waypoints |
| GIS / mapping | Compute polygon attributes |
| Computer graphics | Mesh area, hit detection |
| Robotics | Workspace and obstacle areas |
Limitations
The formula assumes a simple (non-self-intersecting) polygon. For polygons with holes, compute the outer area and subtract the inner areas separately. Curved boundaries must first be approximated by enough straight edges.