**Problem Analysis**

To find the diameter of the protractor, we need to use the information given in the problem. We know that the perimeter of the protractor is 108 cm. However, a protractor is a semicircle, so we need to consider only half of its circumference.

The formula to calculate the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. In this case, we have a semicircle, so we need to divide the circumference by 2.

Let's solve the problem step by step.

**Solution**

1. Let's assume the diameter of the protractor is D cm. Therefore, the radius of the protractor is half of the diameter, which is D/2 cm.

2. The formula to calculate the circumference of a circle is C = 2πr. Since we have a semicircle, we need to divide the circumference by 2. Therefore, the formula becomes C = πr.

3. We are given that the perimeter of the protractor is 108 cm. This is the distance around the semicircle. So, we can substitute C with 108 cm in the formula. We get 108 cm = πr.

4. We want to find the diameter, which is twice the radius. So, we need to find the value of r first. Rearranging the equation from step 3, we get r = 108 cm / π.

5. Now, we can substitute the value of r in the equation for the diameter. The diameter D = 2r. Substituting the value of r from step 4, we get D = 2 * (108 cm / π).

6. To find the numerical value of the diameter, we need to know the value of π. π is approximately equal to 3.14159.

7. Substituting the value of π in the equation from step 5, we get D = 2 * (108 cm / 3.14159).

8. Simplifying the equation, we get D ≈ 68.58 cm.

**Answer**

The diameter of the protractor is approximately 68.58 cm.

Perimeter of semi circular protractor = πr+2r= r(π+2)108=r(36/7)108×7/36 = r21= r so daimeter = 2r = 21×2 = 42