Largest Square inside a hexagon

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Milanu

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Hi, Hoping someone can help with this garden related question. I have a gazebo each side measuring 1700mm. I need to know the largest square that can fit inside it as i am thinking of installing a hot tub inside. Just by trial an error and a tape measure i think the largest square that would fit inside is around 2150mm square. I am guessing there is a way of calculating this but i'm afraid its beyond my elderly mind. Can anyone help?
 
Rigorous geometry yields an inscribed square length of 2155.5 mm

So very nicely guestimated.
 
Milanu said:
Hi, Hoping someone can help with this garden related question. I have a gazebo each side measuring 1700mm. I need to know the largest square that can fit inside it as i am thinking of installing a hot tub inside. Just by trial an error and a tape measure i think the largest square that would fit inside is around 2150mm square. I am guessing there is a way of calculating this but i'm afraid its beyond my elderly mind. Can anyone help?
Click to expand...
1583092582616.png
 
Ahhh... Missed that. By the time I have looked at the thread the title is no longer visible on the screen. But, sure enough, it's there above the top post.
 
Is it somehow obvious that the max square must have two sides that are parallel to two sides of the hexagon? Or, for that matter, that all four corners of the square must touch the hexagon?
 
LCKurtz said:
Is it somehow obvious that the max square must have two sides that are parallel to two sides of the hexagon? Or, for that matter, that all four corners of the square must touch the hexagon?
Click to expand...

You're right that it's not obvious. But playing around in powerpoint a a minute or two can convince you that it's so.
There's no orientation other than the one shown that allows the square to be fully inside the hexagon.

It is likely however that there are rectangles that can be inscribed with a larger area.
 
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