Problem source

By sketching  on the same axes the graphs of  $y=\sin x$ and
$y=x$, show that, for $x>0$:
\begin{questionparts}
\item $x>\sin x\,$;
\item $\dfrac {\sin x} {x} \approx 1$ for small $x$. 
\end{questionparts}
A regular polygon has $n$ sides, and perimeter $P$. 
Show that the area of the polygon is
\[
\displaystyle \frac{P^2} { {4n \tan \l\dfrac{ \pi} { n} \r}} \;.
\]
Show by differentiation (treating $n$ as a continuous variable)
  that the area of the polygon 
increases as $n$ increases with $P$ fixed.
 Show also that, for large $n$, the ratio of the 
area of the polygon to the 
area of the smallest circle which can be drawn around the polygon is approximately $1$.