I have a large backyard tree around which I want to build a regular hexagonal picnic table.
The tree trunk circumference is 96 inches. How can I figure out the minimum dimensions of the center hexagonal hole for the trunk?
Mathematically, this is asking the dimensions of a hexagonal polygon when inscribed by a circle of given circumference.
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If the radius of the inscribed circle is $r$ then the circumference is $c=2\pi r$ while a side of the hexagon is $s=\frac2{\sqrt{3}}r$ so $$s=\frac1{\sqrt{3}\pi}c$$ and with $c=96$ you would get $s \approx 17.643$ while $r \approx 15.279$.
$s$ is also the radius of the circumscribing circle.
The hexagon has circumference $6s \approx 105.86$
The diameter of the tree circle is $\frac{96}{\pi}$ and the radius is half this. Then the radius of the outer circle circumscribing the hexagon is $$\frac{48}{\pi}\times \sec 30=\frac{96}{\pi\sqrt{3}}$$
This is the same as the side of the required hexagon.