A sector of a circle has a diameter of  18
feet and an angle of  5π6
radians. Find the area of the sector.

To find the area of a sector of a circle, we can use the formula for the area of a sector given the radius \( r \) and the angle \( \theta \) in radians:

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

Here, we are given the diameter of the circle, which is 18 feet. Therefore, the radius \( r \) is half of the diameter:

\[ r = \frac{18}{2} = 9 \text{ feet} \]

The angle of the sector is \( \theta = \frac{5\pi}{6} \) radians.

Now, we can substitute these values into the formula:

\[ A = \frac{1}{2} \times 9^2 \times \frac{5\pi}{6} \]

First, calculate \( 9^2 \):

\[ 9^2 = 81 \]

Now, substitute and simplify:

\[ A = \frac{1}{2} \times 81 \times \frac{5\pi}{6} \]
\[ A = \frac{81}{2} \times \frac{5\pi}{6} \]
\[ A = \frac{81 \times 5\pi}{12} \]
\[ A = \frac{405\pi}{12} \]

Simplify the fraction:

\[ A = \frac{135\pi}{4} \]

Therefore, the area of the sector is:

\[ A = \frac{135\pi}{4} \text{ square feet} \]

Would you like more details on any step or have any questions?

Here are 8 related questions for further practice:
1. What is the circumference of the entire circle with a diameter of 18 feet?
2. How would the area of the sector change if the angle were \( \frac{\pi}{3} \) radians?
3. If the angle of the sector were doubled, what would be the new area?
4. How would you find the length of the arc of this sector?
5. What is the area of the entire circle?
6. How does the area of the sector compare to the area of the entire circle?
7. If the radius were 12 feet and the angle remained \( \frac{5\pi}{6} \), what would be the area of the sector?
8. How would you convert the angle from radians to degrees?

**Tip:** Remember that the area of a sector is directly proportional to the angle. If the angle doubles, the area of the sector also doubles.

Learn how to calculate the area of a sector of a circle with a diameter of 18 feet and an angle of 5π/6 radians using the sector area formula. This problem involves circle geometry and radians. Suitable for high school students.

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