If the number of sides of two regular polygons having the same perimeter be n and 2n respectively, prove that their areas are in the ratio
2 cos ( $π$ /n) : [1 + cos ( $π$ /n)].

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Solution

Let p be the perimeter of both the polygons . Then each sides of first polygon is of length (p / n ) and that of the second polygon is (p / 2n) . If $A_{1}, A_{2}$ demote their areas then by
$A_{1} = \frac{1}{4} n . \frac{p^{2}}{n^{2}} c o t \frac{π}{n}$
and $A_{2} = \frac{1}{4} \cdot 2 n \cdot \frac{p^{2}}{4 n^{2}} \cdot c o t \frac{π}{2 n}$
$∴ \frac{A_{1}}{A_{2}} = \frac{2 c o t \frac{π}{n}}{c o t \frac{π}{2 n}} = \frac{2 c o s \frac{π}{n} s i n \frac{π}{2 n}}{s i n \frac{π}{n} c o s \frac{π}{2 n}}$
$= \frac{2 c o s \frac{π}{n} s i n \frac{π}{2 n}}{2 s i n \frac{π}{2 n} c o s \frac{π}{2 n} c o s \frac{π}{2 n}}$
$= \frac{2 c o s \frac{π}{n}}{2 c o s^{2} \frac{π}{2 n}} = \frac{2 c o s \frac{π}{n}}{1 + c o s \frac{π}{2 n}}$

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