Polygon Area Calculator - Area, Perimeter and Centroid from Coordinates (Shoelace)
Polygon Area Calculator
Area of any shape from its corner coordinates
Any Polygon
Triangles, irregular lots, floor plans and map outlines: anything with straight edges and up to 2000 corners.
Centroid and Perimeter
Also get the centre of mass and the length of the outline, plus the direction the corners run.
Warns About Crossings
If edges cross, the tool tells you, because the area is then ambiguous.
Private
Your coordinates are not stored or sent.
How the Shoelace Formula Works
The shoelace formula, also called Gauss’s area formula, finds the area of a polygon from its corner coordinates alone. Multiply each x by the next corner’s y, subtract each y times the next x, add everything up around the outline and halve it. It is named for the criss-cross pattern of the multiplications, which looks like lacing a shoe. The sign of the sum tells you whether the corners run counterclockwise (positive) or clockwise (negative).
The formula is exact for simple polygons, where the edges only meet at the corners. It does not need the shape to be convex. The coordinates must be in a flat Cartesian system with equal units on both axes, such as metres or feet. Latitude and longitude are angles on a sphere, so for large areas they must be converted to a map projection first.
Key Takeaways
- Checked against Shapely: Area, perimeter and centroid match Shapely on hundreds of random polygons.
- Order matters: List the corners around the outline. A different order draws a different shape.
- Related calculators: See also the Area Calculator and the Triangle Calculator.