As the author says in the first paragraph, his goal is to demonstrate Archimedes' method of calculating the area of a circle. He does that by noting that a circle can be approximated by a regular polygon with "n" sides, all vertices on that circle- and the more sides you take, the better the approximation. He calculates the area of the polygon by drawing lines from the center of the circle to each vertex, dividing the polygon into n isosceles triangles. The area of a triangle is "1/2 base times height". In this case, the base of each triangle is the side of the polygon while height is the distance from the center of the circle to the center of the base. For n large, so that each triangle is very "skinny", that is approximately the length of the two equal sides of the triangle, the radius of the circle.
He specifically uses the sine law to calculate the length of the base. Dividing the $2\pi$ radians of a complete circle by n, the vertex angle of each triangle, at the center of the circle, is $\frac{2\pi}{n}$. Since the three angles of any triangle sum to $\pi$ radians, and the two base angles are equal, they are each $\frac{\pi- \frac{2\pi}{n}}{2}= \frac{(n-2)\pi}{2n}$.
