Polygon area from coordinates
Area, perimeter and centroid of a simple polygon from ordered vertices.
Fill in the fields and the result will appear here automatically.
The shoelace formula gives the area of a simple polygon from vertices entered along its outline. Either clockwise or anticlockwise order is valid: direction changes the signed sum but not geometric area. The calculator also gives perimeter and the centroid of a uniform planar region. Concave outlines are allowed; self-crossings, overlapping edges and zero area are excluded.
How it works
Formula and logic
Let wᵢ = xᵢyᵢ₊₁ − xᵢ₊₁yᵢ and 2A_signed = Σwᵢ; area is |A_signed|. Centroid X = Σ(xᵢ+xᵢ₊₁)wᵢ/(6A_signed), with Y analogous. Perimeter sums neighbouring-vertex distances, including the closing edge. Coordinate products and signs are evaluated before rounding so a large origin shift does not destroy a small area.
Example
A 4 by 3 rectangle entered as four corners gives an area of 12 and a perimeter of 14.
Fields and units
- Vertices: x and y per line, in order — common coordinate length unit
How to use
- — Enter one vertex per line: x and y.
- — Follow the outline in order — either direction works, but do not jump across.
- — Do not repeat the first vertex at the end; the outline closes itself.
- — Check the winding if the shape is not what you expected.
- — Separate x and y with whitespace or a semicolon; a decimal comma is allowed inside a coordinate. Page limits are 3–256 distinct vertices and 32768 characters.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: area of a simple closed region from its boundary
- Limitation
- A simple planar outline in Cartesian coordinates. Centroid refers to a uniformly filled region, not the average of vertices or a non-uniform material. Geographic latitude and longitude are not planar lengths.
FAQ
Do I need to repeat the first point at the end?
No: the outline closes automatically. One repeated first point at the very end is accepted and excluded from the vertex count. Other repeated vertices and zero-length edges are rejected.
Does the direction of the outline matter?
Both directions are valid. Reversing the order changes only the winding label; area, perimeter and centroid stay the same. Reordering vertices so edges cross is an error, but clockwise order itself is not.
Can the polygon be non-convex?
Yes, provided the outline is simple, without crossings or overlaps. For a crossing outline the shoelace sum is signed and a filled-region area needs a chosen fill rule; this tool does not present that sum as the area of a simple region.
What units does the result use?
Coordinates and centroid use one length unit you choose; perimeter uses that unit and area its square. Do not mix metres and centimetres between points; units are not detected automatically.
Why is a straight line rejected?
Because three collinear points do not enclose anything. Showing zero would look like a valid answer to an invalid figure.