How can I show that a sequence of regular polygons with $n$ sides becomes more and more like a circle as $n \to \infty$?

Here's one serious approach: Let $f_n\colon [0,2\pi]\to \Bbb R_+$ be the function whose graph, in polar coordinates, is the regular $n$-gon centered at the origin with a vertex at $(1,0)$. Then $(f_n)$ converges uniformly to a constant function mapping any angle to $1$, whose graph is a circle.

We can also look at the limit of the area, a la Archimedes, and the perimeter.

The regular polygon approaches the circle in the following sense:

• All vertices of the polygon are on the circle.

• The maximal distance of the polygon to the circle is given by $2R\sin^2(\frac{\pi}{2n})$, which goes to zero as $n$ goes to $\infty$.

It seems worth emphasizing that "look more and more like circles" admits numerous interpretations. The answers and comments currently visible say that the polygons converge to the circle in several ways: They eventually lie within arbitrarily narrow annuli just within the circle. Their areas converge to the circle's area. Their perimeters converge to the circle's circumference. One could add more; for example, for almost all rays $R$ emanating from the origin, the direction in which the $n$-gon crosses $R$ converges to the direction in which the circle crosses $R$ (namely, perpendicular to $R$). The "almost" here refers to the unpleasantness that a few (countably many) $R$'s pass through a vertex of one of the polygons, so the direction of crossing is undefined there, but even these $R$'s are OK if one uses the average of the directions just to the left and just to the right of $R$. I suspect there are lots of other convergence properties that one could state and prove in this situation. An interesting but non-mathematical question would be to determine which of the many notions of convergence cause people to say that the $n$-gons for large $n$ "look like circles".