What will be the ratio of the volume of a sphere and a cube if the surface areas of the sphere and the cube are equal ?

Explanation

Let the radius of the sphere = r unit
and let the length of each side of the cube = x unit
As per question , ` 4pir^(2) = 6x^(2)`
` rArr (r^(2))/(x^(2)) = 3/(2pi)`
` rArr r/x = (sqrt(3))/(sqrt(2) .sqrt(pi))`
` :. ` ( volume of the sphere ) : (volume of the cube )
` = 4/3 pir^(3) , x^(3) = (4/3pir^(3))/(x^(3)) = 4/3 pi (r/3)^(3)`
` = 4/3 xx pi xx ((sqrt(3))/(sqrt(2).sqrt(pi)))^(3) = 4/3 xx pi xx (3sqrt(3))/(2sqrt(2).pi sqrt(pi))`
` = (sqrt(2) xx sqrt(3))/(sqrt(pi)) = (sqrt(6))/(sqrt(pi)) = sqrt(6) : sqrt(pi) `
Hence the required ratio = `sqrt(6) : sqrt(pi)`