Abstract
Heron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by
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
H. S. M. Coxeter and S. L. Greitzer,Geometry Revisited, The Mathematical Association of America, Washington, DC, 1967.
Torsten Sillke, Private communication.
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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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DOI: https://doi.org/10.1007/BF02574377