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Unusual Polygon

Age 14 to 16 Short Challenge Level:

The diagram shows a polygon ABCDEFG, in which $FG =6$ and $GA=AB=BC=CD=DE=EF$.
Also $BDFG$ is a square.
The area of the whole polygon is exactly twice the area of $BDFG$.
Find the length of the perimeter of the polygon.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas. 

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.