The triangle formed by the points ADB is equilateral, as are the triangles BEC, DCF and ADG. The point ABCD forms a square, so the angles DBE, ECF, FDG and GAD are all equal to

\[360-90-2 \times 60=150\]

degrees.The perimeter of the shape is then

\[4 \times \frac{150}{360} \times 2 \pi \times 6= 20 \pi\]

.