What the largest regular hexagon within a square of side-length 's'? (2008)

So, I'm trying to make a home-brew quick ref card and it seems to look best if it's six-sided. I'm adept at the 'arts & crafts' trick of rounding square corners with a few 'facet' scissors cuts, so I figured a regular hexagon should be just a few well placed cuts on a square.

Then I found this forum thread ... and, as they say, there goes the rest of my afternoon :-P +++ So far my opening hypothesis is that the diagonal of the hexagon (from vertex to opposite vertex) will lie along the diagonal of the square. But how far along and what the resulting sides(hex):sides(sqr) ratio is, well I haven't got that noodled out yet.