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- Q3748369 subject Q7700464.
- Q3748369 subject Q8826767.
- Q3748369 abstract "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, 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.".
- Q3748369 wikiPageWikiLink Q11500.
- Q3748369 wikiPageWikiLink Q120812.
- Q3748369 wikiPageWikiLink Q127840.
- Q3748369 wikiPageWikiLink Q178546.
- Q3748369 wikiPageWikiLink Q191290.
- Q3748369 wikiPageWikiLink Q19821.
- Q3748369 wikiPageWikiLink Q321237.
- Q3748369 wikiPageWikiLink Q36810.
- Q3748369 wikiPageWikiLink Q37555.
- Q3748369 wikiPageWikiLink Q44337.
- Q3748369 wikiPageWikiLink Q626974.
- Q3748369 wikiPageWikiLink Q6722.
- Q3748369 wikiPageWikiLink Q7700464.
- Q3748369 wikiPageWikiLink Q782746.
- Q3748369 wikiPageWikiLink Q816425.
- Q3748369 wikiPageWikiLink Q8366.
- Q3748369 wikiPageWikiLink Q8826767.
- Q3748369 comment "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.".
- Q3748369 label "Shoelace formula".