The area of a regular polygon is $' n'$ sides is where $' r'$ is inradius, $R$ is circumradius and $' a'$ is side of the triangle

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$\frac{n R^{2}}{2} \sin (\frac{2 π}{n})$

$n r^{2} \tan (\frac{π}{n})$

$\frac{n a^{2}}{4} \cot (\frac{π}{n})$

$n R^{2} \tan (\frac{π}{n})$

answer is A, B, C.

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Detailed Solution

Area of polygon nx area of $Δ O B C$

$= \frac{n a^{2}}{4} . \cot \frac{π}{n}$

Now, $a = 2 r \tan \frac{π}{n}$

$\Rightarrow$ Area of polygon $= n r^{2} \tan (\frac{π}{n})$

Also, $a = 2 R \sin \frac{π}{4}$

$= \frac{n R^{2}}{2} \sin (\frac{2 π}{n})$

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