Polygon Area from Coordinates
Area of any simple polygon from its vertex coordinates by the shoelace formula, with perimeter, centroid, bounding box, winding direction and a drawing.
Area, perimeter, centroid and bounds of the polygon you list, each edge length, a drawing of the shape, and a warning when the boundary crosses itself.
Example: (0,0) (4,0) (4,3) (0,3) is a 4 × 3 rectangle: area 12, perimeter 14, centroid (2, 1.5). The pentagon (1,1) (5,1) (6,4) (3,6) (0,4) has area 21 and perimeter 17.54.
The shoelace,
vertex by vertex.
The formula, what else is computed from the same sums, and the one condition it needs.
The formula
Number the vertices around the boundary. Twice the signed area is Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ), the last vertex pairing with the first; the sign tells the direction (positive for counter-clockwise). The name comes from the criss-cross pattern of the products when the coordinates are written in two columns.
Perimeter, centroid, bounds
Each edge is √(Δx² + Δy²); the perimeter is their sum. The centroid of the area (not of the vertices) is Σ (xᵢ + xᵢ₊₁)(xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ) / 6A for each coordinate. The bounding box is simply the extreme coordinates.
The one condition
The boundary must not cross itself: for a self-intersecting figure the formula subtracts one lobe from the other. The page tests every pair of non-adjacent edges and warns. Coordinates are treated as planar — for map coordinates, project to a plane first. A closing vertex that repeats the first is dropped. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Shoelace (Gauss's area) formula A = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)| and the polygon centroid formula (elementary analytic geometry)
Last reviewed 20 September 2026. How results are checked: How we verify.