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# sphere

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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

#1
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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