I just wanted to point this out: the equilateral triangle isn't really the limiting case for a 3-sided figure. It's actually the
curvilinear triangle with a width of 80 in. It has more area, for a given width.
Also, the triangle isn't the only figure for which this works: see
here for an applet that demonstrates the principle for odd numbers of sides, from 3 to 21.
So, who's going to build a robot that fits a curvilinear pentagon, just to annoy the officials?