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The shoelace formula or shoelace algorithm (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple 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. It has applications in surveying and forestry,〔Hans Pretzsch, ''(Forest Dynamics, Growth and Yield: From Measurement to Model )'', Springer, 2009, ISBN 3-540-88306-1, p. 232.〕 among other areas. The formula was described by Meister (1724-1788) in 1769〔.〕 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. The area formula is derived by taking each edge ''AB'', and calculating the (signed) area of triangle ''ABO'' with a vertex at the origin ''O'', by taking the cross-product (which gives the area of a parallelogram) and dividing by 2. As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference triangle remains. This is why the formula is called the Surveyor's Formula, since the "surveyor" is at the origin; if going counterclockwise, positive area is added when going from left to right and negative area is added when going from right to left, from the perspective of the origin. The area formula is valid for any non-self-intersecting (simple) polygon, which can be convex or concave. ==Definition== The formula can be represented by the expression: : where * ''A'' is the area of the polygon, * ''n'' is the number of sides of the polygon, and * (''x''''i'', ''y''''i''), ''i'' = 1, 2,..., ''n'' are the vertices (or "corners") of the polygon. Alternatively:〔〔(Shoelace Theorem ), ''Art of Problem Solving Wiki''.〕 : where ''x''''n''+1 = ''x''1 and ''x''0 = ''x''''n'', as well as ''y''''n''+1 = ''y''1 and ''y''0 = ''y''''n''. If the points are labeled sequentially in the counterclockwise direction, then the above determinants are positive and the absolute value signs can be omitted;〔 if they are labeled in the clockwise direction, the determinants will be negative. This is because the formula can be viewed as a special case of Green's Theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Shoelace formula」の詳細全文を読む スポンサード リンク
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