



 
Hi Paul, I am going to let $d$ be the diameter of the disk you need to turn and $w$ the width of the hexagon, the distance between the flat sides. Thus in my diagram the distance from $C$ to $B$ is $\large \frac{d}{2}$ and the distance from $C$ to $D$ is $\large \frac{w}{2}.$ The measure of the angle $BCA$ is $\large \frac{360}{6} \normalsize = 60$ degrees and hence the measure of the angle $BCD$ is 30 degrees. Hence \[ \cos (30) = \frac{\sqrt{3}}{2} = \frac{CD}{CB} = \frac{w/2}{d/2}\] and thus \[d = \frac{2}{\sqrt{3}} w = 1.1547\; w.\] Penny  


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