The surface area of a cube is equal to that of a sphere. If x is the volume of the cube and y is the volume of the sphere, then what is x^2:y^2 equal to?

Explanation

Given 6a^2 = 4\pi r^2 \implies a^2 = \frac{2\pi}{3}r^2. Volumes are x = a^3 and y = \frac{4}{3}\pi r^3. Therefore, \frac{x^2}{y^2} = \frac{a^6}{\frac{16}{9}\pi^2r^6} = \frac{(\frac{2\pi}{3}r^2)^3}{\frac{16}{9}\pi^2r^6} = \frac{\frac{8\pi^3}{27}}{\frac{16\pi^2}{9}} = \frac{\pi}{6}.

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