Finding The Area of a Circle With Calculus

Question

This is my first attempt to actually go ahead and prove the area of a circle is $\pi r^2$ using the methods shown on calculus. I first used a coordinate system and placed the centre of my circle (with radius $r$) at the origin. Now, we increase the number of sides of the inscribed regular polygon; we note that the number of sides (call it $n$) is the same as the number of isosceles triangle “sectors” that the polygon consists of. If the angle of each of these triangles is $\theta$, then the base of each triangle (or equivalently, one side of the regular polygon) is easily seen to be $r\sin\theta \over \cos\frac{\theta}{2}$. The apothem of the regular polygon is easily seen to be $r\cos\frac{\theta}{2}$. So that the area of each isosceles triangle “sector” is $\frac{1}{2}r^2\sin\theta$. Since there are $n$ triangles, the area of the whole polygon $\frac{1}{2}r^2\sin\theta n$. This can be used as the function take the limit of as it’s variable approaches infinity, however, it has two variables $\theta$ and $n$. So we rewrite $\theta$ as a function of $n$, which is $\theta = –\frac{\pi}{8} n$. Hence our function is $f(n) = – \frac{r^2\sin(\frac{\pi}{8})n}{2}$. This is where I am stumped. I need to find $\lim_{x\to\infty} f(x)$. However, the limit of $\sin x$ as $x$ approaches infinity is undefined. How do you continue from this point on? Thank you in advance.