To find the area of a sector of a circle, we can use the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( A \) is the area of the sector, \( \theta \) is the angle in degrees, and \( r \) is the radius of the circle.
Given:
- \( \theta = 216^\circ \)
- \( r = 7 \) inches
Let's substitute the values into the formula:
\[ A = \frac{216}{360} \times \pi (7)^2 \]
Calculating \( (7)^2 \):
\[ (7)^2 = 49 \]
Now substituting that into the equation:
\[ A = \frac{216}{360} \times \pi \times 49 \]
Next, simplify \( \frac{216}{360} \):
\[ \frac{216}{360} = \frac{216 \div 72}{360 \div 72} = \frac{3}{5} \]
So we can rewrite the area \( A \):
\[ A = \frac{3}{5} \times \pi \times 49 \]
Now multiply:
\[ A = \frac{3 \times 49}{5} \times \pi = \frac{147}{5} \times \pi \]
Thus, the area of the sector is:
\[ \boxed{\frac{147}{5} \pi} \text{ square inches} \]
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