Math team test

[prisma grove]

a sphere is both inscribed in a cube and circumscribed about another cube what is the ratio of the edge of the smaller cube to the edge of the larger cube - answer is root 3 over 3 but how do i get to that plz explain

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[restive garnet]

the radius of the sphere, r, is half the side length of the large cube, and also the distance from the centre of the small cube to one of its vertices

[cloud acorn]

would the problem be equivalent to a problem where we consider a circle that is inscribed in a square and circumscribes about another square?

i think it should be it isn't equivalent, but it is helpful to think of the two-dimensional analogue as a simpler problem

if we think of the problem in terms of a circle between squares, rather than a sphere between cubes, and then rotate the bigger square by a quarter-degree rotation, then we should recover the remark by @restive garnet

[upper tapir]

toe

[cloud acorn]

in the three dimensional analogue, the line "r" is still half the edge of the larger cube. It is also half the diagonal of the cube -- but that diagonal is not the diagonal along the surface of one of the faces of the cube (which is equivalent to the line of length 2r in the two-dimensional analogue), it is instead a diagonal through the body of the cube.