The circumference of a circle with radius $r$ is $2 \pi r$.
Each piece has a quarter of the circle with radius $4$ together with two semicircular arcs of radius $2$.
The arc of the large circle has length $\frac{1}{4} \times 2 \times \pi \times 4 = 2 \pi$.
The semicircular arcs have length $\frac{1}{2} \times 2 \times \pi \times 2 = 2 \pi$.
Therefore the overall perimeter is $3 \times 2\pi = 6\pi$.