Given a regular right hexagonal prism of edge length 1, what is the largest circle (in terms of area) that can be placed completely inside the prism?

Extension:

What is the largest circle (in terms of n) that can be placed completely inside a regular right unit n-prism?

If n=3 you can't do better than putting your circle on one of the square sides of the prism, so the solution is a circle of diameter 1 and an area of π/4

As n increases the prism approaches a cylinder.  In a cylinder you can do no better than the circular base (picture a coin spinning in a tube.)  So the limiting area as n increases is that of the circle that can be inscribed in the base.  r = 1/(2tan(180/n))  A = πr^2

Posted by Jer on 2012-07-01 14:27:19