The derivation of pi using limits

[potmobius]

I was looking up how pi was derived, and found this:

http://mathforum.org/library/drmath/view/52589.html

Question: How were the quantities n*sin(180/n) and n*tan(180/n) found/derived?

 

[Werg22]

The first one is seen from taking the perimeter of a regular n-gon with "radius" (distance from a vertex to the "center") 1, and letting n go to infinity. The idea is that as n grows larger, the perimeter of the n-gon gets closer to the circumference of the unit circle, which is 2pi. The perimeter of the n-gon is 2n*sin(180/n), so n*sin(180/n) must approach pi as n goes to infinity. The second one is seen easily by writing n*tan(180/n) as n*sin(180/n) / cos(180/n), the denominator goes to 1 as n goes to infinity, so its limit as n goes to infinity of n*tan(180/n) is the same as n*sin(180/n).