如图，将底面半径为$1$高为$3$的圆锥截去体积为$\dfrac{\pi }{27}$的锥尖，剩余圆台的侧面积为$(\qquad)$．

$\\dfrac{8\\sqrt{10}\\pi }{9}$

$\\dfrac{16\\sqrt{10}\\pi }{9}$

$\\dfrac{16\\sqrt{2}\\pi }{9}$

$\\dfrac{32\\sqrt{2}\\pi }{9}$

设锥尖的半径为$r$，高为$h$，

$\because $ 锥尖的体积为$\dfrac{\pi }{27}$，

$\therefore \dfrac{\pi }{27}=\dfrac{1}{3}\pi {r}^{2}h$，

解得${r}^{2}h=\dfrac{1}{9}$，

又$\because \dfrac{r}{1}=\dfrac{h}{3}$，

$\therefore h=3r$，

$\therefore r^{2}\cdot 3r=\dfrac{1}{9}$，

解得$r=\dfrac{1}{3}$，

$\therefore h=1$，

$\therefore $ 圆台侧面积$S=\dfrac{1}{2}\times 2\pi \times 1\times \sqrt{1+9}-\dfrac{1}{2}\times 2\pi \times \dfrac{1}{3}\times \sqrt{1+\dfrac{1}{9}}=\dfrac{8\sqrt{10}\pi }{9}$．

故选：$\rm A$