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A regular polygon has $$100$$ sides each of length $$1$$. Another regular polygon has $$200$$ sides each of length $$2$$. When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
The area of a regular polygon with $$n$$ sides of side length $$s$$ is $$\frac{ns^2}{4\tan(\pi/n)}$$. Therefore the ratio of the area of the $$200$$-gon to the $$100$$-gon is $$4\frac{\tan(\pi/100)}{2\tan(\pi/200)}$$, which is approximately $$8$$. Hence the closest listed integer is $$8$$.
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