The sum of the lengths of the $12$ edges of a cuboid is $\quantity{x}{cm}$. The distance from one corner of the cuboid to the furthest corner is $\quantity{y}{cm}$. What in $\mathrm{cm^2}$ is the total surface area of the cuboid?
(A) $\quad \dfrac{x^2-2y^2}{2} \qquad$ (B) $\quad x^2+y^2 \qquad$ (C) $\quad \dfrac{x^2-4y^2}{4} \qquad$ (D) $\quad \dfrac{xy}{6} \qquad$ (E) $\quad \dfrac{x^2-16y^2}{16}$