6. Given that the plane containing $\triangle P A D$ is perpendicular to the plane containing rectangle $A B C D$, $P A=P D=A B=2, \angle A P D=60^{\circ}$, if $P, A, B, C, D$ are all on the same sphere, then the surface area of this sphere is $\qquad$
Given that $\triangle A P D$ is an equilateral triangle, we know that $A B C D$ is a square. From the model of a triangular prism, we have $R^{2}=\left(\frac{2 \sqrt{3}}{3}\right)^{2}+1=\frac{7}{3} \Rightarrow S_{\text {sphere}}=4 \pi R^{2}=\frac{28}{3} \pi$.