Abstract: 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 $a_n$ in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1. In this article, we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of $\{a_n\}$ which corresponds to non arithmetic Veech surfaces is $(5-\sqrt{5})/20$, which is uniquely realized by the area of the minimal triangles of the normalized golden $L$-shaped translation surface up to affine equivalence.

Key words: Veech surfaces, minimal triangles, non arithmetic