The triangles forming the polygon are isosceles. The area of an isosceles triangle is given by
The angle theta, in this case, is . And there are n of these triangles.
So, you have
Of course, as the number of triangles grows larger and larger, the area of the polygon will appraoach the area of the circle. This is similar to the method Archimedes used to find the area of a circle by inscribing a n-sided polygon.
Except, he didn't use trig. As far as we know.
Let's NOT use L'Hopital. No call for laziness.
Of course this famous limit is 1. And we have
Don't forget the