Polygon Area Calculator (Shoelace)

About this calculator

The shoelace formula (also called Gauss's area formula) computes the area of any simple polygon directly from the (x, y) coordinates of its vertices, listed in order around the shape: Area = ½|Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|, summed over all vertex pairs and wrapping back to the first point at the end — the name comes from the crisscross pattern the multiplication terms trace when written out, resembling laced shoe eyelets. Perimeter follows separately by summing the straight-line distance between consecutive vertices.

This calculator takes a list of (x, y) vertex coordinates in order and returns the polygon's area and perimeter. Land surveyors and GIS analysts use this exact formula to compute parcel or plot areas from a set of boundary coordinates, and game developers and computational geometry programmers use it inside collision detection, mesh processing, and procedural generation code wherever an arbitrary polygon's area needs to be known from its vertices alone.