Polygon Area Calculator (Shoelace)

How it is calculated

Enter one x,y pair per line (or separated by semicolons), in order around the outline. For n vertices the engine sums xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ, wrapping the last point back to the first, and returns area = |sum| ÷ 2. The perimeter is the sum of the straight distances √(Δx² + Δy²) between consecutive points, including the closing side. Results are shown to four decimals, in the unit of your coordinates (squared for the area). At least three valid points are required. Taking the absolute value means clockwise or counter-clockwise order both work.

Common mistakes

• Listing the points in the wrong order (for instance jumping between opposite corners); the formula then describes a self-crossing figure and gives a smaller or even zero area. A bowtie such as (0,0), (4,4), (4,0), (0,4) returns 0.0000.
• Repeating the first point at the end "to close" the shape; it still gives the right area and perimeter, but the vertex count is one too high.
• Mixing units, such as some coordinates in metres and others in feet, or using latitude and longitude degrees and expecting square metres.
• Forgetting that the perimeter is in the same unit as the coordinates while the area is in that unit squared.