In this article we obtain an approximation of a given fuzzy number by a unique polygonal fuzzy number preserving its main characteristics such as core, support, and expected interval. For this, we consider the well-known weighted L2 metric. We show that this weighted polygonal approximation is equivalent to a strict convex quadratic optimization problem finite dimensional with linear inequality and equality constraint. Then, we present efficient and practice solution methods illustrated with several examples. Some properties of invariance of the approximation operator are also obtained. The results presented here extend and improve the methods obtained for weighted trapezoidal approximation and piecewise linear approximation of fuzzy numbers.