When I was teaching infinitesimal equivalency between $\sin(x)$ and $x$ ($x\rightarrow0$) for Calculus, I realized that it was not very easy to have a pure elementary proof for it without using the area formula for sectors. Indeed, the area formula for sectors itself can be derived by this equivalency. It seems that we have a circular argument here. A very smart student suggested that we can simply use the fact that among all shapes with a fixed perimeter the disk has the largest area. She further gave a proof sketch for this statement by symmetry and the fact that when one makes two edges of a triangle orthogonal then its area is increased.
I am wondering if there is any other proof of this equivalency with or without using advanced concepts in Calculus? The difficulty here is that this equivalency is so fundamental in Calculus that many other results (e.g., the derivative of $\sin(x)$ is $\cos(x)$) are based on it so cannot be used to prove it. Meanwhile, elementary mathematics is quite unable to deal with curves.