Junior Mathematical Olympiad 2005 Section B Q15

Points $A$, $B$, $C$, $D$, $E$ and $F$ are equally spaced around a circle of radius 1. The circle is divided into six sectors as shown:

The six sectors are then rearranged so that $A$, $B$, $C$, $D$, $E$ and $F$ lie on a new circle, also of radius 1, as shown with the sectors pointing outwards:

Find the (exact) area of the curvy unshaded region.