Suppose we put on the unit circumference n independent uniformly distributed random points and build a convex random polygon with the vertices in these points. What are the average area and the average perimeter of this polygon? The average area was calculated by K. Brown some years ago. We calculatе the average perimeter and obtain quite similar formulae. In the same time we discuss the rate of convergence of this value to the limit. We evaluate also the average value of the sum of squares for the sides of the inscribed triangle.