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

r : s is √3 : 2

Step-by-step explanation:

The given parameter are;

A cube is inscribed in a sphere

The side length of the cube = s

The radius of the sphere = r

The ratio r : e = Required

It is noted that for a cube inscribed in a sphere, we have;

The diameter of the sphere = The diagonal of the cube

The diameter of the sphere, D = 2 × The radius = 2·r

The square of the diagonal of the cube, d² = s² + s² + s² = 3·s²

∴ The diagonal of the cube, d = (√3)·s

From the relationship between the cube and the sphere in which it is inscribed, (The diameter of the sphere = The diagonal of the cube), we have;

2·r = (√3)·s

∴ r/s = (√3)/2

r : s = √3 : 2.