Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence ${a_n}$.And each $a_n$ corresponds to finitely many affine equivalent classes of Veech surfaces with area 1.Smillie and Weiss also gave the first three elements of ${a_n}$,all of which are arithmetic Veech surfaces.In this paper,we give an algorithm for calculating the area of the minimal triangle and prove the area of the minimal triangle of the normalized golden $L$-shaped translation surface is the fourth element of the ordered sequence ${a_n}$ and its affine equivalent class is also the first non arithmetic Veech surfaces corresponding to the the ordered sequence ${a_n}$.