The shape consists of several segments:

1. A straight line of length $4 \, \text{cm}$.
2. A circular arc with radius $2 \, \text{cm}$ subtended by a $120^\circ$ angle.
3. A straight line of length $3 \, \text{cm}$.
4. A circular arc with radius $3 \, \text{cm}$ subtended by a $210^\circ$ angle.

Step 1: Perimeter of straight lines

The total length of the straight segments is: $4 \, \text{cm} + 3 \, \text{cm} = 7 \, \text{cm}.$

Step 2: Perimeter contribution from the arcs

The length of an arc is given by the formula: $\text{Arc length} = 2 \pi r \times \frac{\theta}{360^\circ},$ where $r$ is the radius and $\theta$ is the central angle.

Arc 1:

For the first arc with radius $2 \, \text{cm}$ and angle $120^\circ$: $\text{Arc length} = 2 \pi \times 2 \times \frac{120^\circ}{360^\circ} = \frac{4\pi}{3} \, \text{cm}.$

Arc 2:

For the second arc with radius $3 \, \text{cm}$ and angle $210^\circ$: $\text{Arc length} = 2 \pi \times 3 \times \frac{210^\circ}{360^\circ} = \frac{7\pi}{3} \, \text{cm}.$

Step 3: Total perimeter

Adding the straight lines and arc lengths together: $\text{Perimeter} = 7 + \frac{4\pi}{3} + \frac{7\pi}{3} = 7 + \frac{11\pi}{3} \, \text{cm}.$ Thus, the exact value of the perimeter is: $7 + \frac{11\pi}{3} \, \text{cm}.$

Let me know if you need more details or have any questions!

Here are 5 related questions for you:

1. How do you calculate the length of a circular arc?
2. What is the formula for the circumference of a full circle?
3. How does the angle affect the length of an arc?
4. Can a sector's area be calculated using a similar formula as the arc length?
5. How would the perimeter change if the angles were different?

Tip: For circular arcs, always double-check whether the angle is given in degrees or radians before applying the formula!