Question

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

Answer 1

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