A cyclic polygon is a Polygon with Vertices upon which a Circle can be Circumscribed. Since every
Triangle has a Circumcircle, every Triangle is cyclic. It is conjectured that for a cyclic polygon
of sides, (where is the Area) satisfies a Monic Polynomial of degree , where

(1) | |||

(2) |

(Robbins 1995). It is also conjectured that a cyclic polygon with sides satisfies one of two Polynomials of degree . The first few values of are 1, 7, 38, 187, 874, ... (Sloane's A000531).

For Triangles
, the Polynomial is Heron's Formula, which may be written

(3) |

(4) |

which is of order in . Robbins (1995) gives the corresponding Formulas for the Cyclic Pentagon and Cyclic Hexagon.

**References**

Robbins, D. P. ``Areas of Polygons Inscribed in a Circle.'' *Discr. Comput. Geom.* **12**, 223-236, 1994.

Robbins, D. P. ``Areas of Polygons Inscribed in a Circle.'' *Amer. Math. Monthly* **102**, 523-530, 1995.

Sloane, N. J. A. Sequence A000531 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-25