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)
|
 |  | ![1/2[(2m+1)(2m; m)-2^(2m)]](/images/equations/CyclicPolygon/Inline10.svg) |
(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, ... (OEIS A000531).
For triangles 
, the polynomial
is Heron's formula, which may be written
 |
(3)
|
and which is of order 
in 
.
For a cyclic quadrilateral, the polynomial
is Brahmagupta's formula, which may be written
 |
(4)
|
which is of order 
in 
.
Robbins (1995) gives the corresponding formulas for the
cyclic pentagon and cyclic
hexagon.
For a set of 
side lengths that form a simple closed polygon, there exists a cyclic 
-gon for every 
with these side lengths. Moreover, this cyclic polygon
has the largest possible area among all 
-gons with the same side lengths (Demir 1966, Oxman 2024).
