A hexagon (not necessarily regular) on whose polygon vertices a circle may be circumscribed. Let

(1)

denote the th-order symmetric polynomial on the six variables consisting of the squares of the hexagon side lengths , so

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

Then let be the area of the hexagon and define

			(8)
			(9)
			(10)
			(11)
			(12)

The area of the hexagon then satisfies

(13)

or this equation with replaced by , a seventh-order polynomial in . This is times the polynomial discriminant of the cubic equation

(14)

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.

CITE THIS AS:

Weisstein, Eric W. "Cyclic Hexagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CyclicHexagon.html