What is the limit of the area of an n-sided polygon, as n approaches infinity?

Question

What is the limit of the area of an n-sided polygon, as n approaches infinity?

Is it essentially the same as the area of a circle with radius $r$, i.e $\pi r^2$?

Or am I mistaken?

Answer 1

For a $n$-sided polygon, you can divide the polygon into $n$ triangles by joining center to vertices. You can give its area as:

$$A_n=\frac{n}{2}r^2\sin(\frac{2\pi}{n})$$

where $r$ is distance from center to a vertex.

As $n\to\infty$;

$$\lim_{n\to\infty}A_n=\pi \cdot r^2\cdot\frac{\sin(\frac{2\pi}{n})}{\frac{2\pi}{n}}= \pi r^2$$

Since $\lim_{x\to\infty}\frac{\sin x}{x}=1$.