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Discrete & Computational Geometry

, Volume 12, Issue 2, pp 223–236 | Cite as

Areas of polygons inscribed in a circle

  • D. P. Robbins
Article

Abstract

Heron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by
$$K = \sqrt {s(s - a)(s - b)(s - c)} ,$$
wheres is the semiperimeter (a+b+c)/2. Brahmagupta gave a generalization to quadrilaterals inscribed in a circle. In this paper we derive formulas giving the areas of a pentagon or hexagon inscribed in a circle in terms of their side lengths. While the pentagon and hexagon formulas are complicated, we show that each can be written in a surprisingly compact form related to the formula for the discriminant of a cubic polynomial in one variable.

Keywords

Side Length Discrete Comput Geom Irreducible Polynomial Monic Polynomial Laurent Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    H. S. M. Coxeter and S. L. Greitzer,Geometry Revisited, The Mathematical Association of America, Washington, DC, 1967.zbMATHGoogle Scholar
  2. 2.
    Torsten Sillke, Private communication.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • D. P. Robbins
    • 1
  1. 1.Center for Communications ResearchInstitute for Defense AnalysesPrincetonUSA

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