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. For a cyclic polygon with
sides, where 
, 
(with 
the signed area) satisfies
a monic polynomial of degree 
whose coefficients
are polynomials with integer coefficients in the squared side lengths. Here
 |  |  |
(1)
|
 |  | ![1/2[(2m+1)(2m; m)-2^(2m)]](/images/equations/CyclicPolygon/Inline11.svg) |
(2)
|
(Robbins 1995). For 
sides, the corresponding monic polynomial has
degree 
and factors into two polynomials
of degree 
when the product of the side lengths is adjoined to
the ring of polynomials in
the squared side lengths. These statements, originally conjectured by Robbins (1995),
have been proved (Fedorchuk and Pak 2005, Maley et al. 2005). 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 convex cyclic
quadrilateral, a linear factor gives 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 convex
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).
