Geometry And Shapes Codexery

Polygon

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A polygon is a geometric figure with multiple angles and sides.[1][2] The word comes from Greek roots meaning "many" and "corner" or "angle".[1] Ancient civilizations were already familiar with polygons.[3]

Quick Facts

Field
Geometry

Facts from the source article.

Key facts and history

In 1952, Geoffrey Colin Shephard generalized polygons to the complex plane.[4] The interior angles of regular star polygons were first studied by Poinsot.[2] Polygons may be characterized by their convexity or type of non-convexity.[5] The idea of a polygon has been generalized in various ways, including spherical and skew polygons.[6]

The ancient Greeks were aware of regular polygons; a pentagram (a star polygon) is depicted on a 7th-century B.C. krater.[3] Thomas Bradwardine, during the 14th century, conducted the earliest documented systematic investigation of non-convex polygons.[7] A polygon's number of corners always equals its number of sides.[2] For a simple polygon with n sides, the total of its interior angles equals (n − 2) multiplied by 180 degrees.[2]

This result follows because any simple n-gon can be divided into (n − 2) triangles, each contributing 180 degrees to the sum.[2]

What the research sources say

The Math Open Reference derivation shows a regular polygon can be broken into n congruent isosceles triangles.[8] The area of a regular polygon can be expressed using the radius R of its circumscribed circle.[9] The signed area of a simple polygon can be computed using the shoelace formula.[10][11] The signed area depends on the ordering of vertices and orientation of the plane.[11]

The area of a self-intersecting polygon can be defined in two different ways.[12] When vertices are listed in counterclockwise order, the computed signed area yields a positive value.[11]

References

  1. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Craig, "A new universal etymological technological, and pronouncing dictionary of the English language", Oxford University, 1849, p. 404 (original not read by us) en.wikipedia.org
  2. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Kappraff, "Beyond measure: a guided tour through nature, myth, and number", World Scientific, 2002, p. 258 (we also read that source) en.wikipedia.org
  3. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Heath, "A History of Greek Mathematics, Volume 1", Courier Dover Publications, 1981, p. 162; Cratere with the blinding of Polyphemus and a naval battle , Castellani Halls, Capitoline Museum, accessed 2013-11-11. Two pentagrams are visible near the cente (we also read that source) en.wikipedia.org
  4. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Shephard, G.C.; "Regular complex polytopes", Proc. London Math. Soc. Series 3 Volume 2, 1952, pp 82–97 (original not read by us) en.wikipedia.org
  5. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Magnus, "Noneuclidean tesselations and their groups", Academic Press, 1974, p. 37 (original not read by us) en.wikipedia.org
  6. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Coxeter (3rd Ed 1973); Günter Ziegler (1995). "Lectures on Polytopes". Springer Graduate Texts in Mathematics, . p. 4. (original not read by us) en.wikipedia.org
  7. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Coxeter, H.S.M.; Regular Polytopes, 3rd Edn, Dover (pbk), 1973, p. 114 (original not read by us) en.wikipedia.org
  8. Area of a regular polygon – derivation from Math Open Reference.. facts only, no text reused. mathopenref.com
  9. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Area of a regular polygon – derivation from Math Open Reference.; A regular polygon with an infinite number of sides is a circle: limn→+∞R2·n2·sin2πn=π·R2 (we also read that source) en.wikipedia.org
  10. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: B.Sz. Nagy, L. Rédey: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949); Bourke, "Calculating The Area And Centroid Of A Polygon", July 1988 (original not read by us) en.wikipedia.org
  11. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: Bart Braden, "The Surveyor's Area Formula", The College Mathematics Journal, 1986, p. 326–337 (original not read by us) en.wikipedia.org
  12. Wikipedia, "Polygon" (CC BY-SA 4.0), which cites: De Villiers, "Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral", Learning and Teaching Mathematics, January 2015, p. 23–28 (original not read by us) en.wikipedia.org

Each numbered sentence above was checked against the source it cites before publication. Sentences citing Wikipedia adapt text from the English Wikipedia article Polygon (contributors), used under CC BY-SA 4.0; this entry is available under the same licence. Other sources were used for facts only; no text or images are copied from them.

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Sources

Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.

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