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
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(1)
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Plugging in together with
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(2)
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(3)
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and eliminating 
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) |
(4)
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This function is plotted above.
Maximizing gives the area of the hexagon as the second-largest real root of
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(5)
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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)
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so the optimal solution is 3.9% larger.
The values of 
and 
corresponding to the maximal solution are given by
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(7)
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(8)
|
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(9)
|
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(10)
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