Irregular Polygon Area Calculator — Shoelace Formula
Enter the (x, y) coordinates of each vertex of your polygon — in order around the boundary — and get the area and perimeter instantly using the Shoelace formula. Works for any simple polygon regardless of shape.
Computed by the Shoelace (Gauss) formula — works for any simple (non-self-intersecting) polygon
How does this calculator work?
The Shoelace formula A = ½|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| gives the exact area of any simple polygon from its vertex coordinates listed in boundary order. Enter x,y pairs (semicolon-separated), and the perimeter is computed from the summed edge lengths.
Formula
How this is calculated
The Shoelace formula (also called the Gauss area formula or surveyor's formula) computes the area of any simple polygon from the Cartesian coordinates of its vertices listed in order — either clockwise or counter-clockwise. You pair up consecutive vertices (xᵢ, yᵢ) and (xᵢ₊₁, yᵢ₊₁), sum the cross-products xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ around the loop, take the absolute value, and divide by 2. The same formula that gives a positive area for CCW order gives a negative area for CW order, so the absolute value handles both.
The perimeter is calculated by summing the Euclidean distances between consecutive vertices, including the closing edge from the last vertex back to the first. Both quantities use the same coordinate system you enter your points in, so if your coordinates are in metres the area is in square metres.
The formula assumes a simple polygon — one whose edges do not cross each other. If you enter vertices that create a self-intersecting (butterfly) shape the result will be the net signed area rather than the total enclosed area, which may not be what you intend. Keep vertices in consistent order around the boundary for a correct result.
Frequently asked questions
Yes — vertices must be listed in order around the polygon boundary (either all clockwise or all counter-clockwise). The Shoelace formula sums cross-products around the loop, so a wrong ordering (jumping across the shape) produces an incorrect result. The calculator takes the absolute value, so CW and CCW both give the correct positive area.
Yes. The Shoelace formula works correctly for any simple polygon, whether convex or concave, as long as the edges do not cross each other. Just list the vertices in consistent boundary order.
The area is in the square of whatever unit your coordinates are in — if you enter metres, the area is in square metres. If you want the result in a different unit, multiply or divide accordingly.
Also known as
TG we-Calculate Editorial Team. (2026). Irregular Polygon Area Calculator — Shoelace Formula [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/irregular-polygon-area-calculator
TG we-Calculate Editorial Team. "Irregular Polygon Area Calculator — Shoelace Formula." TG we-Calculate. 2026. https://we-calculate.com/calculator/irregular-polygon-area-calculator.
TG we-Calculate Editorial Team, "Irregular Polygon Area Calculator — Shoelace Formula," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/irregular-polygon-area-calculator
@misc{wecalculate_irregular_polygon_area_calculator, title = {Irregular Polygon Area Calculator — Shoelace Formula}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/irregular-polygon-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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