Question

A sphere and a cube have equal surface areas. Show that the ratio of the volume of sphere to that of cube is root6 : rootπ

Answer 1

Heya !!!


Let the radius of the sphere be R and the edge of the cube be A.


Then,


Surface area of the sphere = Surface area of the cube


=> 4πR² = 6A²



=> A² = 2/3 πR²



=> A = ✓2/3 πR²



=> A = R ✓2π/3




Therefore,



Required ratio =


Volume of Sphere : Volume or cube



=> 4/3 πR³ : A³


=> 4/3 πR³ : (2/3) πR³ × ✓2π/3




=> 2 : ✓2π/3


=> ✓2 : ✓π/3



=> ✓6 : ✓π..... PROVED......


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