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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.

 Jul 17, 2022
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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 \({4 \pi r^2 \over 24r^2 } = {4 \pi \over 24} = {\color{brown}\boxed{ \pi \over 6}}\)

 Jul 17, 2022

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