Let a be the side of the circumscribed pentagon and b be that of the inscribed pentagon.
First note that for the circle is inscribed for the exterior pentagon and circumscribed for the interior pentagon. Therefore the in−radius of the exterior polygon, say r is equal to the circum−radius, say R of the interior pentagon, i.e., R=r. See the figure below.
Using standard formula, a=2rtanπ5,b=2Rsinπ5
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Area of a regular polygon having n sides is n×(side)24cotπn.
Therefore the required ratio is (5×b24cotπ55×a24cotπ5)=b2a2=(2Rsinπ5)2(2rtanπ5)2=cos2π5

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