AREA

Let P be a polygon on a convex surface. This polygon can be divided into arbitrarily small triangles, and, therefore, there exists a sequence Zn of its partitions such that the maximal diameter of triangles in the nth partition tends to zero as n→∞. To each triangle of the nth partition, we put in correspondence the plane triangle with sides of the same length and take the sum SZn of the areas of these plane triangles. It turns out that these sums SZn converge to some limit S as n→∞ independently of the choice of the partitions Zn whenever these partitions are infinitely refined. This limit we define to be the area of the polygon P .