shoelaceformelen

The shoelaceformelen, commonly known in English as the shoelace formula or Gauss’s area formula, is a method for finding the area of a simple polygon when the coordinates of its vertices are known. If a polygon has vertices (x0, y0), (x1, y1), ..., (x_{n-1}, y_{n-1}) taken in either clockwise or counterclockwise order, and we let (x_n, y_n) = (x0, y0), the area A is given by A = 1/2 times the absolute value of the sum over i of (x_i y_{i+1} − x_{i+1} y_i).

The intuition behind the formula is that it accumulates the oriented area of trapezoids formed by each edge and the x-axis, which, when combined, yields the polygon’s total area. The terms pair up to produce a cancellation that depends only on the vertex coordinates and their order.

The formula requires that the polygon be simple (its edges do not cross) and that the vertices be listed in a consistent order (either all clockwise or all counterclockwise). If the absolute value is not taken, the formula returns a signed area whose sign indicates orientation. For polygons with holes, the outer boundary and inner boundaries can be handled by summing their areas with appropriate signs.

The shoelaceformula is widely used in geographic information systems, computer graphics, and computational geometry because it is simple, robust, and runs in linear time with respect to the number of vertices. It also highlights that the area can be computed purely from vertex coordinates without explicitly triangulating the polygon.