SOLUTION: what is the 2 column proof for theorem 11.8 (circumference of a circle)?

Question 409794: what is the 2 column proof for theorem 11.8 (circumference of a circle)?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Proving the circumference of a circle is is a little tricky using a two-column proof. It's also difficult to construct a proof of the circumference without using calculus.

The best way is to consider a regular n-sided polygon and its perimeter. Suppose the distance from the center of an n-sided polygon to one of its vertices is (where is constant). We can construct isosceles triangles, each one looking something like this:

where . Hence, the base of the isosceles triangle is (applying the definition of sine). We have n of these segments so the perimeter of the n-gon is

. Taking the limit as goes to infinity,

. Let . This limit is equivalent to

. Applying L'Hopital's rule this is equivalent to

, which is the circumference of the circle.
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