Find the area of the sector of a circle with diameter 32 feet and an angle of 6𝜋/7 radians. BUT DO NOT ROUND UNTIL THE FINAL ANSWER (Round your answer to four decimal places.)

To find the area of the sector of a circle, we use the formula:

\[
\text{Area of Sector} = \frac{1}{2} r^2 \theta
\]

Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in radians.

Given:
- The diameter of the circle is 32 feet, so the radius \( r \) is:

\[
r = \frac{32}{2} = 16 \text{ feet}
\]

- The angle \( \theta = \frac{6\pi}{7} \) radians.

Now, substitute the values into the formula:

\[
\text{Area of Sector} = \frac{1}{2} \times 16^2 \times \frac{6\pi}{7}
\]

Calculate the area step-by-step:

1. First, calculate \( 16^2 \):

\[
16^2 = 256
\]

2. Multiply by \( \frac{1}{2} \):

\[
\frac{1}{2} \times 256 = 128
\]

3. Multiply by \( \frac{6\pi}{7} \):

\[
\text{Area of Sector} = 128 \times \frac{6\pi}{7} = \frac{768\pi}{7} \text{ square feet}
\]

Now, compute the exact value:

\[
\text{Area of Sector} = \frac{768 \times 3.141592653589793}{7} 
\]

Finally, rounding to four decimal places:

\[
\text{Area of Sector} \approx 344.8321 \text{ square feet}
\]

So, the area of the sector is **344.8321 square feet**.

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. What would the area of the sector be if the angle were doubled?
2. How would the area change if the diameter were halved?
3. What is the area of the entire circle with a 32 feet diameter?
4. How does the sector's area compare if the angle is \( \frac{\pi}{2} \) radians?
5. What would be the area if the radius were increased by 50% but the angle remained \( \frac{6\pi}{7} \)?

**Tip:** Always ensure the angle is in radians when using the sector area formula. If given in degrees, convert it first!

Learn how to calculate the area of a sector of a circle with a diameter of 32 feet and an angle of 6𝜋/7 radians using the area of sector formula. This problem involves circle geometry and sector area calculations in advanced high school mathematics.

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