Question 1108536: A cube of side length s sits inside a sphere of radius r so that the vertices of the cube sit on the sphere. Find the ratio r : s. Found 2 solutions by KMST, ikleyn:Answer by KMST(5328) (Show Source):
You can put this solution on YOUR website! Let us pretend that this sketch really looks like the cube
The seven visible vertices are labeled A through G.
Vertex H is in back. In the attempted perspective view,
H is directly behind D, and directly below E. is the length of a side of the cube. is the length of the diagonal of a cube face.
A cross-section (cutting through A, F, G, and D, would look like this is the length of the diagonal of the cube,
the longest possible distance between two points of the cube.
For the cube to fit in the sphere, that has to be the diameter of the sphere.
So, --> .
The ratio r:s is (about 0.866).
If "s" is the cube's side length, then the cube's longest 3D diagonal is = .
At the same time, this longest diagonal is the DIAMETER of the sphere:
= 2r.
Therefore, = .
