Cube circumscribed to a sphere of radius 5

Qwertyuiop[]

Qwertyuiop[]

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A Cube is Circumscribed To A Sphere of Radius 5. We can say that:
a) The diagonal of one face of the cube measures 10 sqrt(2)
b) A diagonal of the cube measures 10 sqrt(2)
c) It's impossible for a cube to be circumscribed to a sphere
d) One side of the cube measures 10 / sqrt (2)
e) One side of the cube measures 10 sqrt(2)

So c) is obviously wrong. I had 2 attempts at this question, i will show my working for both. I made a wrong assumption in the first attempt.

Attempt 1 ) I will attach a pic named attempt 1 to show what i did. So i made 1 wrong assumption here . The length of the legs of the triangle have length 5. I got it wrong completely. The diagonal of the face of the cube is not 5 . 5 is the radius of the sphere which is the same as the circumradius, half the length of the diagonal of the cube. I think it's because my drawing is in 2d and in a hurry i labelled the face diagonal as being of the same length as the half the body diagonal of the cube. I now understand why I had it wrong the first time so tried it again. Also is this a right angle triangle ?
EDIT : I got 5 sqrt(2) for my first attempt. This is option d) as 10 / sqrt(2) rationalizes to 5 sqrt (2).

Attempt 2) I need help for this 2nd attempt because although I think my working is correct , the answer is not listed ??

I get 2 lengths , the length of a side of the cube and the diagonal length of a face.(working attached below)

I get (10* sqrt(6)) / 3 as face diagonal length and this option is not listed. I checked on a few website calculators that will find the lengths of sides, diagonals or radius of sphere given one length. I used this : https://calcresource.com/solid-cube.html . For a diagonal of length 10, the edge length is 5.7735 which is same as 10/sqrt3. And it gives 5 as circumradius which the radius of our sphere! I also confirmed the answer with another website calculator: https://www.volumeofcube.com/circumscribed-sphere-radius. I don't understand what I did wrong , what is the correct answer then ? Thank you.
 

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Qwertyuiop[] said:
A Cube is Circumscribed To A Sphere of Radius 5. We can say that:
a) The diagonal of one face of the cube measures 10 sqrt(2)
b) A diagonal of the cube measures 10 sqrt(2)
c) It's impossible for a cube to be circumscribed to a sphere
d) One side of the cube measures 10 / sqrt (2)
e) One side of the cube measures 10 sqrt(2)
...
I get 2 lengths , the length of a side of the cube and the diagonal length of a face.(working attached below)

I get (10* sqrt(6)) / 3 as face diagonal length and this option is not listed. I checked on a few website calculators that will find the lengths of sides, diagonals or radius of sphere given one length. I used this : https://calcresource.com/solid-cube.html . For a diagonal of length 10, the edge length is 5.7735 which is same as 10/sqrt3. And it gives 5 as circumradius which the radius of our sphere! I also confirmed the answer with another website calculator: https://www.volumeofcube.com/circumscribed-sphere-radius. I don't understand what I did wrong , what is the correct answer then ? Thank you.
Click to expand...
You're right that your first picture and work are wrong.

Your second picture and work are correct (if, as has been pointed out, we change the word to "inscribed"). But then the provided answers would be wrong.

If we take the wording seriously, though, so that the cube is outside the sphere, then the diameter of the sphere is equal to the side of the cube, so that the cube has side 10. Then you will get one of the answers they give!

So the problem is that you are misinterpreting "circumscribed". See here, where a sphere is circumscribed about a cube, and here, where we are told "To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F"." The trouble is that we usually see spheres circumscribed about a cube (as in the sites you used), so it's hard to even find an example of a cube circumscribed about a sphere, and the wording is tricky, especially where your problem says "circumscribed to", which is not standard in my experience, and easily confused with "circumscribed by".

I think the main point of the problem is to read carefully!
 
Dr.Peterson said:
You're right that your first picture and work are wrong.

Your second picture and work are correct (if, as has been pointed out, we change the word to "inscribed"). But then the provided answers would be wrong.

If we take the wording seriously, though, so that the cube is outside the sphere, then the diameter of the sphere is equal to the side of the cube, so that the cube has side 10. Then you will get one of the answers they give!

So the problem is that you are misinterpreting "circumscribed". See here, where a sphere is circumscribed about a cube, and here, where we are told "To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F"." The trouble is that we usually see spheres circumscribed about a cube (as in the sites you used), so it's hard to even find an example of a cube circumscribed about a sphere, and the wording is tricky, especially where your problem says "circumscribed to", which is not standard in my experience, and easily confused with "circumscribed by".

I think the main point of the problem is to read carefully!
Click to expand...
Thanks for checking my working. I was misinterpreting circumscribed and inscribed, you cleared it out. thank you ?
 
Harry_the_cat said:
circumscribe = draw (a figure) round another, touching it at points but not cutting it.

I agree that "circumscribed TO" is a bit confusing.
Click to expand...
So a cube circumscribed to a sphere would mean , by this definition, a cube drawn round a sphere. Alright! You've all been very helpful thank you!
 
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