https://file.aiquickdraw.com/mathbot/file/1725598665205jhxm4mnp.jpg

To solve for the length \( s \) and area \( A \) of the sector shown in the image, let's go step-by-step:

### 1. Given Information:
- The radius \( r \) of the circle is \( 6 \) units.
- The central angle \( \theta \) of the sector is \( 2\pi/3 \) radians.

### 2. Formula for Length \( s \) of the Arc:
The length of the arc \( s \) of a sector can be found using the formula:

\[
s = r \cdot \theta
\]

### 3. Formula for Area \( A \) of the Sector:
The area \( A \) of the sector can be found using the formula:

\[
A = \frac{1}{2} \cdot r^2 \cdot \theta
\]

### Calculations:

- **Length of the arc \( s \)**:
\[
s = 6 \cdot \frac{2\pi}{3} = 4\pi \, \text{units}
\]

- **Area of the sector \( A \)**:
\[
A = \frac{1}{2} \cdot 6^2 \cdot \frac{2\pi}{3} = \frac{1}{2} \cdot 36 \cdot \frac{2\pi}{3} = 12\pi \, \text{square units}
\]

### Summary of Results:
- Length of the arc \( s = 4\pi \approx 12.57 \) units.
- Area of the sector \( A = 12\pi \approx 37.7 \) square units.

Would you like more details on these steps or further clarifications?

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Here are five related questions to explore further:
1. How does the length of the arc change if the angle is doubled?
2. What would happen to the area if the radius of the circle is halved?
3. How do the formulas for arc length and area change if the angle is in degrees?
4. Can you find the circumference of the full circle based on this information?
5. What is the difference between radian measure and degree measure for angles?

**Tip**: Always ensure your angle is in radians when using the arc length and sector area formulas for consistent results!

Find the length s and area a of a sector given the radius is 6 units and the central angle is 2π/3 radians.

Learn how to find the arc length and area of a sector with a radius of 6 units and a central angle of 2π/3 radians. This problem uses key geometry formulas, including the arc length formula and sector area formula. Suitable for high school students (Grades 9-12).

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