A biggest little polygon with sides is a convex polygon
in the plane of unit polygon
diameter having largest possible area. The biggest little
polygons for ,
8, and 10 are illustrated above. In the figures, diagonals
shown in red have unit length.
Reinhardt (1922) showed that for an odd number, the regular
polygon on
sides is the biggest little -gon. For , the square with diagonal
1 has maximum area, but an infinite number of other quadrilaterals
are equally large (Audet et al. 2002). The case was solved by Graham (1975) and is known as Graham's
biggest little hexagon, and the case was solved by Audet et al. (2002).
Audet, C. "Optimisation globale structurée: propriétés, équivalences et résolution." Thèse
de Doctorat. Montréal, Canada: École Polytechnique de Montréal,
1997. https://www.gerad.ca/Charles.Audet/.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.; and Svrtan, D. "A New Algorithm for Finding Largest Small Polygons Using
Symbolic Computations." 2018. https://crocodays.grad.hr/pres_2018/2018_svrtan.pdf.Audet,
C.; Hansen, P.; and Svrtan, D. "Using Symbolic Calculations to Determine Largest
Small Polygons." J. Global Opt.81, 261-268, 2021. https://doi.org/10.1007/s10898-020-00908-w.Foster,
J. and Szabó, T. "Diameter Graphs of Polygons and the Proof of a Conjecture
of Graham." J. Combin. Th. Ser. A114, 1515-1525, 2007. https://doi.org/10.1016/j.jcta.2007.02.006.Graham,
R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A18,
165-170, 1975. https://doi.org/10.1016/0097-3165(75)90004-7.Hurst,
G. "A Closed Form Expression for the Area of the 'Biggest Little' Octagon and
Decagon." Apr. 14, 2025. https://community.wolfram.com/t/25370.Mulansky,
B. and Potschka, A. "A Zonogon Approach for Computing Small Polygons of Maximum
Perimeter." 2 Apr 2024. https://arxiv.org/abs/2404.01841.Reinhardt,
K. "Extremale Polygone gegebenen Durchmessers." Jahresber. Deutsch.
Math. Verein31, 251-270, 1922.Trela, D. "Maximum-Area
Small Polygons of Even Order." 19 Aug 2026. https://arxiv.org/abs/2608.15666v2.Sloane,
N. J. A. Sequences A111969, A111970,
A111971, A381252,
and A383173 in "The On-Line Encyclopedia
of Integer Sequences."
sides is a convex polygon
in the plane of unit polygon
diameter having largest possible area. The biggest little
polygons for 
,
8, and 10 are illustrated above. In the figures, diagonals
shown in red have unit length.
, the regular
polygon on 
sides is the biggest little 
-gon. For 
, the square with diagonal
1 has maximum area, but an infinite number of other quadrilaterals
are equally large (Audet et al. 2002). The 
case was solved by Graham (1975) and is known as Graham's
biggest little hexagon, and the 
case was solved by Audet et al. (2002).
, Foster and Szabó (2007) proved that the diameter
graph of a biggest little 
-gon consists of a cycle graph
on 
vertices with one pendant
edge attached. This determines which pairs of vertices
are at unit distance, but does not by itself determine
the polygon coordinates or its maximal area.
"BiggestLittlePolygon",
n
]
for 
,
8, and 10.
