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Polygon Area Calculator (Shoelace Formula)

Computes the area of a simple polygon from its vertex coordinates with the shoelace (Gauss) formula, plus its perimeter, centroid, orientation and whether it is convex.

When to use

You have the (x, y) coordinates of a polygon's vertices in order around its boundary and need its area, centroid or perimeter, e.g. a land parcel, floor plan or map shape.

Do not use when: The polygon is regular and you know only its side length (use regular-polygon), it is a triangle given by side lengths (use triangle-area), or the edges cross each other (the shoelace formula is not valid for self-intersecting polygons).

Formula

A = ½ |Σᵢ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| (indices wrap around); perimeter = Σ √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²); centroid = (Σ (xᵢ + xᵢ₊₁)·cᵢ, Σ (yᵢ + yᵢ₊₁)·cᵢ) / (6·A_signed) with cᵢ = xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ

Inputs

Parameter	Type	Unit	Required	Description
`points`	string_list		yes	Vertices as x,y pairs in boundary order (clockwise or counter-clockwise), at least 3, e.g. 0,0 4,0 4,3 0,3. A repeated first point at the end is ignored.

Outputs

Output	Type	Description
`area`	number	\|Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)\| / 2 in the coordinate unit squared.
`perimeter`	number	Sum of the edge lengths, including the closing edge.
`centroid_x`	number	x of the area centroid, Σ (xᵢ + xᵢ₊₁)(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) / (6·A).
`centroid_y`	number	y of the area centroid.
`centroid`	string	The centroid as a coordinate pair, e.g. '(2, 1.5)'.
`is_convex`	boolean	true when every turn along the boundary is in the same direction.
`orientation`	string	'counter-clockwise' (positive signed area) or 'clockwise'.
`vertex_count`	integer	Vertices used after dropping a repeated closing point.

Example

Rectangle 4 × 3: {"points":["0,0","4,0","4,3","0,3"]} → {"area":12,"perimeter":14,"centroid_x":2,"centroid_y":1.5,"centroid":"(2, 1.5)","is_convex":true,"orientation":"counter-clockwise","vertex_count":4}

Pentagon (3,4) (5,11) (12,8) (9,5) (5,6): {"points":["3,4","5,11","12,8","9,5","5,6"]} → {"area":30,"perimeter":26.090056,"centroid_x":7.166667,"centroid_y":7.611111,"is_convex":false,"orientation":"clockwise","vertex_count":5}

GET https://tttkmbb.com/api/v1/calculate/polygon-area?points=0%2C0%2C4%2C0%2C4%2C3%2C0%2C3

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/polygon-area (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/polygon-area · Markdown: https://tttkmbb.com/math/polygon-area.md · JSON definition: https://tttkmbb.com/math/polygon-area.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="polygon-area"
OpenAPI operationId: calculate_polygon_area
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Shoelace formula (reference)
Wikipedia – Centroid – of a polygon (reference)
Wolfram MathWorld – Polygon Area (reference)

FAQ

Does the vertex order matter?

The vertices must follow the boundary (either direction); listing them in a different order describes a different, possibly self-intersecting, polygon. Clockwise input only changes the sign of the signed area, which is reported as orientation.

Can I use latitude/longitude?

Only for small areas after projecting to metres (e.g. UTM); degrees of longitude are not the same length as degrees of latitude, so raw coordinates give wrong areas.

Related calculators

Regular Polygon Calculator — Regular polygons from side length or radius.
Triangle Area Calculator — Triangle area from sides, base and height or coordinates.
Distance Between Two Points Calculator — Length of a single edge between two points.