Two equal radius, but maximally sized, spheres are tangent inside a cube. What is the ratio of r (radius of a sphere) to d (edge of the cube)?

Answer 1

Don't know why the answer didn't show up, it was an answered question.

The spheres are stacked diagonally in 3-D inside the cube. Sphere radius = r, cube edge = d.

You must equate two different paths from corner to corner of the cube, through the centers of the spheres. Imagine a smaller cube, nestled into the corner of the large cube, with one corner coincident with the large cube corner, and the opposite corner coincident with the center of the nearer sphere. Then the diagonal of that small cube, which can be traversed as 3 orthogonal segments of length r each, is (by Pythagorean theorem) of length √3r . Continuing on the large cube diagonal further, the sphere1-to-sphere2 center-to-center distance is 2r, and the remaining distance to the far corner is another √3r , That's 2(1+√3)r in all. Or, by traversing the large cube edges, the same diagonal is √3d. Therefore d = (2/√3)(1+√3)r =( 2+2√3/3) * r .

-- Cheers, -- Mr. d. (the submitter)

Answer 2

2r=d

cube=d^3