Polygon Area from Coordinates Calculator
Enter the ordered x,y vertices of any simple polygon to find its area, perimeter and centroid using the shoelace (surveyor) formula.
From 4 vertices (shoelace formula)
How does this calculator work?
Paste your polygon vertices as x,y pairs in order and this tool applies the shoelace formula: area = ½ of the absolute cross-product sum, plus the perimeter from edge lengths and the centroid from the signed-area weighted average. Works for any simple polygon, no repeated closing point needed.
Formula
How this is calculated
List the polygon vertices as "x,y", one per line, walking around the boundary in order (clockwise or counter-clockwise). At least three vertices are required, and the edge from the last vertex back to the first is added automatically, so do not repeat the starting point.
The signed area is A = ½ Σ (xi·yi+1 − xi+1·yi), summed cyclically over all edges. The polygon area is the absolute value |A|; the sign of A tells you the winding direction (positive for counter-clockwise). The perimeter is the sum of edge lengths √(Δx² + Δy²). The centroid is Cx = 1/(6A) Σ (xi + xi+1)(xi·yi+1 − xi+1·yi) and Cy = 1/(6A) Σ (yi + yi+1)(xi·yi+1 − xi+1·yi), using the signed A.
Units follow the coordinates: if x and y are in metres, area is in square metres and perimeter in metres. The method assumes a simple (non-self-intersecting) polygon; self-crossing shapes give a net signed area that may not match the visual region, and a degenerate set of collinear points yields zero area and is rejected.
Frequently asked questions
Yes. The shoelace formula traces the boundary edge by edge, so list vertices sequentially around the perimeter. Either clockwise or counter-clockwise works; the area magnitude is the same.
No. The calculator automatically closes the polygon by connecting the last vertex back to the first, so just list each corner once.
The formula still returns a signed sum, but for self-crossing shapes that value may not equal the visible enclosed area. The shoelace method is intended for simple polygons.
Also known as
TG we-Calculate Editorial Team. (2026). Polygon Area from Coordinates Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polygon-area-shoelace-calculator
TG we-Calculate Editorial Team. "Polygon Area from Coordinates Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/polygon-area-shoelace-calculator.
TG we-Calculate Editorial Team, "Polygon Area from Coordinates Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polygon-area-shoelace-calculator
@misc{wecalculate_polygon_area_shoelace_calculator, title = {Polygon Area from Coordinates Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polygon-area-shoelace-calculator}}, year = {2026}, note = {TG we-Calculate} }
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