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Areas of polygons inscribed in a circle

  • Published: 06 September 2005
  • Volume 12, pages 223–236, (1994)
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Areas of polygons inscribed in a circle
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  • D. P. Robbins1 
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  • 32 Citations

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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.

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References

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

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  2. Torsten Sillke, Private communication.

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Author information

Authors and Affiliations

  1. Center for Communications Research, Institute for Defense Analyses, 08540, Princeton, NJ, USA

    D. P. Robbins

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  1. D. P. Robbins
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Cite this article

Robbins, D.P. Areas of polygons inscribed in a circle. Discrete Comput Geom 12, 223–236 (1994). https://doi.org/10.1007/BF02574377

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  • Received: 03 March 1993

  • Revised: 14 October 1993

  • Published: 06 September 2005

  • Issue Date: July 1994

  • DOI: https://doi.org/10.1007/BF02574377

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Keywords

  • Side Length
  • Discrete Comput Geom
  • Irreducible Polynomial
  • Monic Polynomial
  • Laurent Polynomial
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