Discrete & Computational Geometry

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

Areas of polygons inscribed in a circle

  • D. P. Robbins


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.


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  1. 1.
    H. S. M. Coxeter and S. L. Greitzer,Geometry Revisited, The Mathematical Association of America, Washington, DC, 1967.MATHGoogle 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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