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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.

Answers(1)

Answer

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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