Shoelace formula

The shoelace formula, also known as Gauss's area formula and the surveyor's formula, is a mathematical <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> to determine the <a href="/facts/Area/KsNG1G9t">area</a> of a <a href="/facts/Simple_polygon/C5CfrPWf">simple polygon</a> whose vertices are described by their <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> in the plane. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces. It has applications in surveying and forestry, among other areas.
The formula was described by Albrecht Ludwig Friedrich Meister (1724–1788) in 1769 and is based on the trapezoid formula which was described by <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a> and <a href="/facts/Carl_Gustav_Jacob_Jacobi/4Syeauo5">C.G.J. Jacobi</a>. The triangle form of the area formula can be considered to be a special case of <a href="/facts/Green%2527s_theorem/WBDk654A">Green's theorem</a>.
The area formula can also be applied to self-overlapping polygons since the meaning of area is still clear even though self-overlapping polygons are not generally <a href="/facts/Simple_polygon/C5CfrPWf">simple</a>. Furthermore, a self-overlapping polygon can have multiple "interpretations" but the Shoelace formula can be used to show that the polygon's area is the same regardless of the interpretation.

Shoelace formula open-in-new

Shoelace formula