Largest circle inscribed in a cube

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Martin K

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Hey guys!
Need a help with a tiny problem we are supposed to solve and I seriously have no clue what to do with it.
Problem:
Find the largest inscribed circle you can fit into a cube of side a=1.
Any help is much appreciated.
Best regards,
Martin
 
Martin K said:
Hey guys!
Need a help with a tiny problem we are supposed to solve and I seriously have no clue what to do with it.
Problem:
Find the largest inscribed circle you can fit into a cube of side a=1.
Any help is much appreciated.
Best regards,
Martin
Click to expand...

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment. Tell us (show us) exactly where you are stuck. Otherwise we have to start with "definitions".

Can you calculate the diameter of the largest inscribed circle that can be fitted into a SQUARE of side a=1?
 
Subhotosh Khan said:
Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment. Tell us (show us) exactly where you are stuck. Otherwise we have to start with "definitions".

Can you calculate the diameter of the largest inscribed circle that can be fitted into a SQUARE of side a=1?
Click to expand...
Thank you, I have read those :)
I don't know how to calculate it at all. I am stuck at blank paper, that is why I tried to ask here for help. I can imagine somewhat inclined circle in a cube to maximalize the diameter of the circle but I don't know how to calculate it.
To rephrase the question then: Can you calculate the diameter of the largest inscribed circle that can be fitted into a CUBE of side a=1?
Sorry for the bad English. It is not my first language.
Best regards,
Martin
 
Suppose the problem were to find the largest circle that could be inscribed in a square of side length 1. Could you do that?
 
HallsofIvy said:
Suppose the problem were to find the largest circle that could be inscribed in a square of side length 1. Could you do that?
Click to expand...
In that case the diameter of the circle would be equal to the side length of the square, therefore 1?
 
Martin K said:
In that case the diameter of the circle would be equal to the side length of the square, therefore 1?
Click to expand...
Now think -

What would be the side length of the largest square that you can draw inside a cube of side "a"?
 
Some points on this circle would touch the cube. Exactly where would they touch? If I am not missing something this would answer your question.
 
The hard part here is that the circle can be on any plane through the cube, not necessarily one parallel to a face. This makes it hard to visualize, and easy to get wrong.

I have a guess: There is a plane that cuts through a cube forming a regular hexagon; the circle inscribed in this hexagon may well be the answer. Its diameter is greater than the side of the cube, contrary to one's first impression of the problem.

I haven't tried proving that there is nothing larger.
 
I just tried this in a 3d modelling package. It seems very likely that Dr. Peterson is correct because any small rotation away from the "hexagon plane" he describes causes the proposed circle to poke out of the cube. I also could not find a better solution further away from this plane. Screenshot below...

circleInCube.png

FYI: To find this plane I rotated a cube by 45 around the z axis, and then by 54.74 (atan(sqrt(2))) around the x axis. The circle diameter given by Subhotosh Khan touches the cube nicely.
 
Nice pictures!

Another approach to drawing it (and also to calculating the radius) would be to define a plane by two of the vertices of the hexagon, together with the center, and intersect that with the cube.
 
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