Qwertyuiop[]
Junior Member
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- Jun 1, 2022
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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.
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.