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.

Let

Of course this famous limit is 1. And we have

Don't forget the

Multiplying,