Graham's biggest little hexagon is the largest possible (not necessarily regular) convex hexagon with polygon
diameter 1 (i.e., for which no two of the vertices are more than unit distance
apart). It is therefore the biggest little polygon
for the case . The solution is given by the above figure (which is shown
with incorrect proportions in Conway and Guy 1996, p. 207), in which the red
lines are all diagonals of unit length.
To find the hexagon, set up coordinates as illustrated above, then the polygon
area formula gives
(1)
Plugging in together with
(2)
(3)
and eliminating and then gives the formula for as
(4)
This function is plotted above.
Maximizing gives the area of the hexagon as the second-largest
real root of
(5)
approximately given by (OEIS A111969).
Note that the sign of the is positive, not negative as erroneously given in Conway
and Guy (1996). Also compare this with the area of the regular hexagon of diameter
1 (and therefore having circumradius 1/2), which is given by
(6)
so the optimal solution is 3.9% larger.
The values of and corresponding to the maximal solution are given by
Audet, C.; Hansen, P.; Messine, F.; and Xiong, J. "The Largest Small Octagon." J. Combin. Th. Ser. A98, 46-59, 2002.Audet,
C.; Hansen, P.; Messine, F.; and Perron, S. "The Minimum Diameter Octagon with
Unit-Length Sides: Vincze's Wife's Octagon is Suboptimal." J. Combin. Th.
Ser. A108, 63-75, 2004.Conway, J. H. and Guy, R. K.
"Graham's Biggest Little Hexagon." In The
Book of Numbers. New York: Springer-Verlag, pp. 206-207, 1996.Foster,
J. abd Szaba, T. "Diameter Graphs of Polygons and the Proof of a Conjecture
of Graham." J. Combin. Th., Ser. A114, 1515-1525, 2007.Graham,
R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A18,
165-170, 1975.Hurst, G. "A Closed Form Expression for the Area
of the 'Biggest Little' Octagon and Decagon." Apr. 14, 2025. https://community.wolfram.com/groups/-/m/t/3444306.Klein,
A. and Wessler, M. "The Largest Small Polytopes." 19 Dec 2002. http://arxiv.org/abs/math.CO/0212262.Sloane,
N. J. A. Sequences A111969, A111970,
and A111971 in "The On-Line Encyclopedia
of Integer Sequences."Trott, M. The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 46-47,
2006. http://www.mathematicaguidebooks.org/.
. The solution is given by the above figure (which is shown
with incorrect proportions in Conway and Guy 1996, p. 207), in which the red
lines are all diagonals of unit length.
and 
then gives the formula for 
as
![A(x)=1/2[sqrt(1-x^2)+x(2-2sqrt(1-x^2)+sqrt(3-4x(1+x)))].](/images/equations/GrahamsBiggestLittleHexagon/NumberedEquation2.svg)
(OEIS A111969).
Note that the sign of the 
is positive, not negative as erroneously given in Conway
and Guy (1996). Also compare this with the area of the regular hexagon of diameter
1 (and therefore having circumradius 1/2), which is given by
and 
corresponding to the maximal solution are given by
