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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 âˆš6 : âˆšÏ€

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 âˆš6 : âˆšÏ€.

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 4Ï€r2 = 6a2 (r/a)2 = 3 / 2Ï€ r / a = âˆš(3/2Ï€) Volume of sphere / Volume of cube = (4/3)Ï€r3 / a3 = (4Ï€/3)(r/a)3 = (4Ï€/3)(âˆš(3/2Ï€))3 = (4Ï€/3)(3/2Ï€)(âˆš(3/2Ï€)) = 2âˆš(3/2Ï€) = âˆš(4x3/2Ï€) = âˆš(6/Ï€) Thus, Volume of sphere : Volume of cube = âˆš6 : âˆšÏ€.
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