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)
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(2)
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(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, ... (OEIS A000531).
For triangles , the polynomial is Heron's formula, which may be written
(3)
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and which is of order in . For a cyclic quadrilateral, the polynomial is Brahmagupta's formula, which may be written
(4)
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which is of order in . Robbins (1995) gives the corresponding formulas for the cyclic pentagon and cyclic hexagon.