Polygons circumscribed to a circle with the smallest circumference

Question

Let $n \geq 3$. Prove that among all $n$-polygons circumscribed to a (particular) circle, the regular $n$-polygon has the smallest circumference.

It shouldn't be difficult, but I don't even know how to start.

Answer 1

For a polygon circumscribed to a circle, the area is half the product between the circle's radius and the polygon's perimeter, since the radius is perpendicular on any tangent, and all the polygon's sides are tangents to the circle $($and, of course, we all know the formula for a triangle's area$)$. Now, the extrema of a function are to be found amongst the roots of its first order derivative. I trust that you can take it from here.