inscribed circle in $n$-gon

Question

If I'm given a circle with radius $r$ and I want to create a polygon with side $n$ (say $n=5$) which can cover the circle fully, then how to prove that a regular polygon is the solution with minimum area?

Answer 1

If you believe without proof that a minimal area circumscribed $n$-gon exists you can argue as follows:

Let $A$, $B$, $C$ be three successive points of tangency with $\angle(AOB)=\alpha$, $\angle(BOC)=\beta$. Then $\alpha+\beta<\pi$, and the polygonal area contained in the sector $AOC$ of central angle $\alpha+\beta$ is given by $$\tan{\alpha\over2}+\tan{\beta\over2}={\sin{\alpha+\beta\over2}\over\cos{\alpha\over2}\cos{\beta\over2}}={2\sin{\alpha+\beta\over2}\over\cos{\alpha+\beta\over2}+\cos{\alpha-\beta\over2}}\ .$$ Keeping $A$ and $C$ (and with them $\alpha+\beta$) fixed this can be increased if $\alpha\ne\beta$. It follows that the smallest area $n$-gon has to be regular.