Some days my brain just doesn't feel like working and this is one of them. I have a problem. I want to make a hexagon that would fit exactly within a square for whatever reason, whether it be for a logo, or for an engineering problem. Doesn't matter.
Now I know from intuition that the hexagon must be oriented at a 45 degree (or 1/4 radian) angle. I just want to know where I should be putting the points that hexagon would be inscribed into, in traditional cartesian coordinates, based on my square's side length of A.
Now that I know what I want to do, how would I find those positions and prove that they're where my hexagon's points should be located?
This question should be easy, but for the life of me I can't figure it out. The solutions I came across so far detail the area of the hexagon in question, and among the proofs for area I observed, I figured out that the points intersecting the square boundary should be
corner + sidelength / (1 + sqrt(3))
However, the trigonometry required to find the points for the vertices not intersecting the square border around the shape still eludes me.
Best Answer
Maybe these are what you are looking for:
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