To find the area of the 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 central angle in degrees, and \( r \) is the radius of the circle.
- Find the radius: The diameter of the circle is given as 18 inches, so the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{18}{2} = 9 \text{ inches} \]
- Substituting values into the formula: Now substitute \( \theta = 122^\circ \) and \( r = 9 \) inches into the formula:
\[ A = \frac{122}{360} \times \pi (9)^2 \]
- Calculate \( r^2 \):
\[ (9)^2 = 81 \]
- Substituting this back into the formula:
\[ A = \frac{122}{360} \times \pi \times 81 \]
- Calculating the fraction:
\[ \frac{122}{360} = \frac{61}{180} \]
- Finding the area:
\[ A = \frac{61}{180} \times \pi \times 81 = \frac{61 \times 81 \pi}{180} \]
- Calculating \( 61 \times 81 \):
\[ 61 \times 81 = 4941 \]
- Now substituting back to find the area:
\[ A = \frac{4941 \pi}{180} \]
- Calculating the numerical value of the area: To find the numerical value of \( \frac{4941 \pi}{180} \):
\[ \frac{4941}{180} \approx 27.005555 \]
- Finding the area in terms of \(\pi\):
\[ A \approx 27.01 \pi \]
Thus, the area of the sector is:
\[ \boxed{27.01 \pi} \text{ in}^2 \]
To two decimal places, this is \( A \approx 27.01 \pi \) in².