Someone showed me a derivation for the area of a circle today. They took a circle of radius $r$ and inscribed a regular polygon in the circle. If you take an $n$-sided polygon, then its area is:
If you let $n$ go to infinity, then you get $\pi$$r^2$ as your area.
However, you are using the limit for $\sin x/x$ as $x$ goes to $0$ to derive this. In order to derive that limit, you need to show that $\sin x<x<\tan x$, which is done using the unit circle and comparing the areas of two triangles and a sector. To find the area of the sector, you need to know the area of a circle. Almost appropriately, we've reached a circular logic.
Is there any way around this?