Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The shoelace formula, or shoelace algorithm, is a mathematical algorithm to determine the area of a polygon whose vertices are described by ordered pairs in the plane. The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces. It is also sometimes called the shoelace method. In mathematics, an ordered pair is a collection of objects having two coordinates (or entries or projections), such that one can always uniquely determine the object, which is the first coordinate (or first entry or left projection) of the pair as well as the second coordinate (or second entry or right projection). If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b. Cartesian products and binary relations (and hence the ubiquitous functions) are defined in terms of ordered pairs.