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.
Best Answer
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 $6s \approx 105.86$