https://file.aiquickdraw.com/mathbot/file/1725815449713s4mkue7v.jpg

The problem provided in the image reads as follows:

**Problem Statement:**
"As shown in the figure below, the radius of the base of a cone is \( r \). Its side expansion is a sector with a central angle of \( 216^\circ \), and the radius \( R \) of the sector is \( 20 \) cm. Find the value of the radius \( r \) of the base (in cm)."

**Analysis and Solution:**

The side expansion of the cone is represented by a sector of a circle. The key relationship to use here involves the arc length of the sector and the circumference of the base of the cone.

1. **Circumference of the base of the cone:**  
   The circumference of the base of the cone, \( C \), is given by:
   \[
   C = 2\pi r
   \]

2. **Arc length of the sector:**  
   The arc length of the sector, \( L \), which corresponds to the circumference of the base of the cone, can be calculated using the formula for the arc length:
   \[
   L = \frac{\theta}{360^\circ} \times 2\pi R
   \]
   where \( \theta = 216^\circ \) and \( R = 20 \) cm.

   Substituting the values:
   \[
   L = \frac{216^\circ}{360^\circ} \times 2\pi \times 20
   \]
   Simplifying:
   \[
   L = \frac{216}{360} \times 40\pi = \frac{3}{5} \times 40\pi = 24\pi \text{ cm}
   \]

3. **Equating arc length to circumference:**
   Since the arc length of the sector corresponds to the circumference of the base of the cone:
   \[
   2\pi r = 24\pi
   \]
   Simplifying:
   \[
   r = 12 \text{ cm}
   \]

Thus, the radius \( r \) of the base of the cone is **12 cm**.

---

Would you like further details on any part of the solution? Here are some related questions you might explore:

1. How would the solution change if the central angle was different?
2. What is the significance of the sector's radius in this problem?
3. How is the relationship between the sector and the cone's base circumference derived?
4. Can this method be applied to solve problems involving different shapes?
5. What would happen if the radius of the sector was doubled?

**Tip:** Understanding how geometric properties are related, such as the connection between a sector's arc and a cone's base, can help solve a variety of problems involving curved surfaces.

As shown in the figure below, the radius of the base of a cone is r. Its side expansion is a sector with a central angle of 216°, and the radius of the sector is 20 cm. Find the value of the radius r of the base (in cm).

Solve a geometry problem involving the side expansion of a cone as a sector with a central angle of 216°. Learn how to calculate the base radius using arc length and circumference formulas. Ideal for students in Grades 9-11.

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