The difference between the areas of the circu | vedclass

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Question

(A) For a regular polygon with $n$ sides and side length $a$,the circumradius $R$ and inradius $r$ are given by:$R = \frac{a}{2} \csc\left( \frac{\pi}

Vedclass · Accepted Answer

$\pi \, cm^2$

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(A) For a regular polygon with $n$ sides and side length $a$,the circumradius $R$ and inradius $r$ are given by:$R = \frac{a}{2} \csc\left( \frac{\pi}{n} ight)$ and $r = \frac{a}{2} \cot\left( \frac{\pi}{n} ight)$.Given $a = 2 \, cm$ and $n = 10$ (decagon):$R = \frac{2}{2} \csc\left( \frac{\pi}{10} ight) = \csc\left( \frac{\pi}{10} ight)$$r = \frac{2}{2} \cot\left( \frac{\pi}{10} ight) = \cot\left( \frac{\pi}{10} ight)$The difference in areas is $\pi R^2 - \pi r^2 = \pi (R^2 - r^2)$.Using the identity $\csc^2 	heta - \cot^2 	heta = 1$:$\pi (\csc^2(\frac{\pi}{10}) - \cot^2(\frac{\pi}{10})) = \pi (1) = \pi \, cm^2$.

The difference between the areas of the circumcircle and the incircle of a regular decagon of side $2 \, cm$ is -

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