Recently, the author and two other researchers constructed a calculation method for the Dunkl-Williams constant $DW(X)$ of a normed linear space $X$. Using the method, we determined the constant of $\mathbb{R}^2$ with symmetric octagonal norms. In this paper, we calculate the Dunkl-Williams constant of its dual space. As the result, the space $\mathbb{R}^2$ with symmetric octagonal norm becomes an example for which the Dunkl-Williams constant of the own space and the dual space have same value.