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# A sphere and a cube have the same surface area . show that the ratio of the volumes of the sphere to that of the cube is √6 : √π

A sphere and a cube have the same surface area . show that the ratio of the volumes of the sphere to that of the cube is √6 : √π.

Raghunath Reddy

Member since Apr 11, 2014

Let r and a be the radius of the sphere and edge of the cube respectively. Given, Surface area of sphere = Surface area of cube 4πr2 = 6a2 (r/a)2 = 3 / 2π r / a = √(3/2π) Volume of sphere / Volume of cube = (4/3)πr3 / a3 = (4π/3)(r/a)3 = (4π/3)(√(3/2π))3 = (4π/3)(3/2π)(√(3/2π)) = 2√(3/2π) = √(4x3/2π) = √(6/π) Thus, Volume of sphere : Volume of cube = √6 : √π.
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