A sphere is inscribed in a cube. What is the ratio of the surface area of the inscribed sphere to the surface area of the cube? Express your answer as a common fraction in terms of pi.

Guest Jul 17, 2022

#1**0 **

Let the radius of the sphere be \(r\).

The surface area of the sphere is \(4 \pi r^2\).

Note that the side length of the cube is the same as the diameter of the sphere, or \(2r\).

This means that the surface area is \((2r)^2 \times 6 = 24r^2\)

So, the ratio is \({4 \pi r^2 \over 24r^2 } = {4 \pi \over 24} = {\color{brown}\boxed{ \pi \over 6}}\)

BuilderBoi Jul 17, 2022