jmc

The sum of the angle measures of a hexagon is $180(6-2)=720$ degrees, so each angle of a regular hexagon measures $720^{\circ }/6=120^{\circ }$. Therefore, $\angle BAF=120^{\circ }$, which means $\angle FAP=180^{\circ }-\angle BAF=60^{\circ }$. Similarly, $\angle PFA=60^{\circ }$. Since the angles of $△APF$ sum to $180^{\circ }$, we have $\angle APF=180^{\circ }-60^{\circ }-60^{\circ }=\boxed{60^{\circ }}$.