Step 1: Here $R=3$ cm (outer radius), $r=2$ cm (inner radius) and $\theta=60^{\circ}$.
Step 2: The boundary of the shaded region consists of: the outer arc, the inner arc and two straight pieces each of length $R-r$.
Step 3: Length of the outer arc
$$=\frac{\theta}{360^{\circ}}\times 2\pi R=\frac{60}{360}\times 2\pi(3)=\pi\text{ cm}$$
Step 4: Length of the inner arc
$$=\frac{60}{360}\times 2\pi(2)=\frac{2\pi}{3}\text{ cm}$$
Step 5: The two straight portions together
$$=2(R-r)=2(3-2)=2\text{ cm}$$
Step 6: Total perimeter
$$=\pi+\frac{2\pi}{3}+2=\frac{5\pi}{3}+2$$
Step 7: Substitute $\pi=\dfrac{22}{7}$.
$$=\frac{5}{3}\times\frac{22}{7}+2=\frac{110}{21}+2=5.24+2=7.24\text{ cm}$$
Hence the perimeter of the shaded region is approximately $7.24$ cm.