Question

Surface area of a cube and sphere are equal. Show that their volumes are in the ratio √π : √6

Answer 1

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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

${4\pi r}^{2} = {6a}^{2}$


(r/a)^2= 3 / 2π

r / a = √(3/2π)

Volume of sphere / Volume of cube = (4/3)πr^3/ a^3 = (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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