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What is the ratio of the areas of the regular pentagons inscribed inside and circumscribed around a given circle?

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    I just provided an answer on another question you posted here. Can I ask what the source of these questions is? If it's homework, you need to show what you tried and let us help from there. – 2017-01-15
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    Re: tag edit: A circle is 2D, not a sphere which is 3D. similarly, A regular pentagon is 2D. And so I changed your two tags referencing spheres. – 2017-01-15

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First of all, both Pentagon's are similar, so it is enough to find the ratio between the distance from the center to the vertex.

For the inside Pentagon that distance is the radius $r$.

For the outside pentagon we have the distance $d$ given by

$$ \cos 36° = \frac{r}{d}$$

So the ratio between the area is equal to the square of the similarity ratio:

$$\left(\frac{r}{d}\right)^2=( \cos 36°)^2$$