Polygon Signed Area Centroid

Description

Compute algebraic signed polygon area and shoelace centroid, returning null for zero algebraic area; caller supplies a simple ordered polygon, no topology validation.

Planar geometry operates on two-coordinate Cartesian points and explicit shape boundaries. A closed rectangle includes its edges, while a segment parameter locates a point along a supporting line. Signed cross products, areas and axis gaps preserve orientation or direction information that a magnitude alone would lose.

Compute algebraic signed polygon area and shoelace centroid, returning null for zero algebraic area; caller supplies a simple ordered polygon, no topology validation.

When to use this tool

Use this tool to compute algebraic signed polygon area and shoelace centroid, returning null for zero algebraic area; caller supplies a simple ordered polygon, no topology validation.

How to use the tool and how it works

Enter points using the input definitions below.
Run the operation. Resolve malformed-input errors before interpreting the result.
Read result.signedArea, result.centroid. Interpret signs, units, ordering and null values using the rules below.

Close the ordered vertex cycle implicitly. Sum v = xj×yNext−xNext×yj, return signedArea = sum(v)/2, and centroid = sum((point+next)×v)/(3×sum(v)); centroid is null when the algebraic sum is zero.^[1]

Input field	Meaning and units	Accepted input
points	Ordered planar [x,y] points. Polygon operations close the cycle implicitly; polyline operations use adjacent pairs only.	Dense list of at most 128 rows of at most 128 finite numbers; the operation’s shape constraints also apply.

Output field	Interpretation
result	Object containing the fields below.
result.signedArea	Algebraic area in squared coordinate units; sign follows vertex winding.
result.centroid	Shoelace centroid [x,y], or null for exactly zero algebraic area.

json

{
  "input": {
    "points": [
      [
        0,
        0
      ],
      [
        2,
        0
      ],
      [
        0,
        2
      ]
    ]
  },
  "output": {
    "result": {
      "signedArea": 2,
      "centroid": [
        0.6666666666666666,
        0.6666666666666666
      ]
    }
  }
}

Limitations and assumptions

The caller must provide suitable ordered polygon vertices. Self-intersection, holes and near-zero area are not diagnosed; area has squared coordinate units.
Point components and rectangle bounds are finite within ±1,000,000,000. Rectangles use [minX,minY,maxX,maxY] with ordered inclusive bounds. Floating arithmetic is used without an implicit robustness tolerance.
Every declared top-level input is required and unknown fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
Numeric outputs must be finite and within the absolute safe-integer magnitude; exact larger integers use decimal text. Source positions are zero-based UTF-16 offsets and span ends are exclusive unless a different convention is explicitly stated.

Alternative or Complementary methods

Choose the boundary and coordinate convention explicitly. Use robust geometric predicates or a topology-aware library when near-coincidence, holes, invalid polygons or geographic coordinates matter.

References

Tool behavior and example: Polygon Signed Area Centroid — tool.io

Similar or alternative tools

Polygon Edge Orientation Table
Report directed polygon edge displacements and radian orientations; zero edges have null direction, and the cycle is implicitly closed.
Polygon Half Plane Clip
Clip an ordered polygon against the closed half-plane nx*x+ny*y<=maximum; no self-intersection validation or geographic geometry.