To find the area of a sector of a circle, we can use the formula:
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.
First, we need to determine the radius of the circle. The diameter is given as 18 inches, so the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{18}{2} = 9 \text{ inches} \]
Now, we can substitute the values into the formula. The angle \( \theta \) is 122°, and the radius \( r \) is 9 inches:
\[ A = \frac{122}{360} \times \pi (9^2) \]
Calculating \( 9^2 \):
\[ 9^2 = 81 \]
Now plugging this in:
\[ A = \frac{122}{360} \times \pi \times 81 \]
Calculating \( \frac{122}{360} \):
\[ \frac{122}{360} \approx 0.338888 \]
Now we calculate the area:
\[ A \approx 0.338888 \times \pi \times 81 \]
Calculating \( 81 \times 0.338888 \):
\[ 81 \times 0.338888 \approx 27.5 \]
Now, substituting this back in for the area:
\[ A \approx 27.5\pi \]
Now, we need to express it in terms of \(\pi\) and convert it to a decimal for the final answer.
Calculating \( 27.5\pi \):
\[ A \approx 27.5 \times 3.14159 \approx 86.493 \]
Rounding to two decimal places, we have:
\[ A \approx 86.49 \]
So, the area of the sector is:
\[ A = 27.5\pi \text{ in}^2 \quad (or \approx 86.49 \text{ in}^2) \]
In \(\pi\) form, the answer is:
\[ A = 27.5\pi \text{ in}^2 \]
In approximate numerical form to two decimal places:
\[ A \approx 86.49 \text{ in}^2 \]
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