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To solve question number 51, let's analyze the problem step by step.

### Problem Analysis:
- A sector of a circle is formed with the center at \( O \) and radius \( OA = OB = 18 \) cm.
- The reflex angle \( \angle AOB = 210^\circ \).
- This sector is to be used to create a cone.
- We need to find the radius of the base of the cone.

### Key Concepts:
1. **Arc Length**: The arc of the sector becomes the circumference of the base of the cone when it is formed. The formula for the arc length \( s \) of a sector is:
   \[
   s = \theta \times r
   \]
   where \( \theta \) is in radians, and \( r \) is the radius of the circle. Since the angle is in degrees, we need to convert it to radians by multiplying by \( \frac{\pi}{180} \).

2. **Circumference of Cone Base**: The arc length \( s \) of the sector will be the circumference of the base of the cone. The circumference \( C \) of the cone's base is given by:
   \[
   C = 2\pi r_{\text{cone}}
   \]
   where \( r_{\text{cone}} \) is the radius of the base of the cone.

3. **Relation**: The arc length of the sector is equal to the circumference of the cone's base:
   \[
   s = C
   \]

### Step-by-Step Calculation:
1. **Convert the Angle to Radians**:
   \[
   \theta = 210^\circ = \frac{210 \times \pi}{180} = \frac{7\pi}{6} \, \text{radians}
   \]

2. **Calculate the Arc Length** (which will be the base circumference of the cone):
   \[
   s = \theta \times r = \frac{7\pi}{6} \times 18 = 21\pi \, \text{cm}
   \]

3. **Set the Arc Length Equal to the Base Circumference**:
   \[
   21\pi = 2\pi r_{\text{cone}}
   \]

4. **Solve for \( r_{\text{cone}} \)**:
   \[
   r_{\text{cone}} = \frac{21\pi}{2\pi} = \frac{21}{2} = 10.5 \, \text{cm}
   \]

Thus, the radius of the base of the cone is **10.5 cm**, which corresponds to option D.

Would you like further details on any of the steps?

### Related Questions:
1. How would the result change if the reflex angle was \( 240^\circ \)?
2. What is the slant height of the cone formed from this sector?
3. How do you calculate the surface area of the cone in this case?
4. What happens if the radius of the sector is increased to 20 cm?
5. How would you calculate the volume of the cone formed?

**Tip**: Always remember that the arc length of a sector becomes the circumference of the cone's base in problems like this.

Jika besar sudut refleks AOB = 210° dan panjang OA = 18 cm, maka panjang jari-jari kerucut yang terjadi adalah ...

Solve a geometry problem where a circle sector with a 210° reflex angle and a radius of 18 cm forms a cone. Learn how to calculate the radius of the cone's base using the arc length and circumference relationship. Suitable for Grades 10-12.

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