I am trying to find the side length of a regular hexagon inscribed within a 12"X12" square. I am not trying to find the maximum side length – I've already seen the formula for that elsewhere – I am trying to find the specific side length where two edges ('east' and 'west', for visualization) are lying on two opposing sides of the square, and the points connecting the other edges ('north' and 'south') are lying in the center of the other two sides of the square.
At first glance, this looks like it should be simple – if I'm working with a regular hexagon, the inner angles should all be 120 degrees. That, coupled with the shape and positioning, means that the 'trimmed' space in the square ought to consist of four 30-60-90 triangles (30 at the 'north' and 'south' markers, 60 at the 'east' and 'west' markers, and 90 at the corners of the square). This doesn't seem to compute, though, because the sides of the triangle along the 'north' and 'south' edges should be 6 inches – but by the formula for a 30-60-90 triangle, the length of the square would have to be 4x in size (x for the first short triangle leg, 2x for the edge of the hexagon, which must equal the hypotenuse of the triangle since it's a regular hexagon, and then another x for the short leg of the other triangle). On a 12" side, this means that x would have to equal three, which blows up the formula for the triangle, as it requires that the x(sqrt(3)) leg be equal in length to the 2x leg.
I'm obviously doing something in my calculations wrong here – where am I thinking incorrectly, and what is the correct answer?
Best Answer
If I've understood the question correctly, the setup is impossible: it assumes that the distances between two opposite vertices of the hexagon, and two opposite sides of the hexagon, are equal to the side length of the square.
Imagining the hexagon inscribed in a circle makes it obvious that opposite vertices are further apart than opposite sides, so the two distances can't be equal.
So what you've got isn't a regular hexagon, and it's refusing to behave like one.