To find the area of a sector, you 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.
In this case, you have \( \theta = 135 \) degrees and \( r = 13 \) inches. Plugging in these values:
\[ A = \frac{135}{360} \times \pi \times (13)^2 \]
First, calculate \( (13)^2 \):
\[ (13)^2 = 169 \]
Now substitute this value back into the area formula:
\[ A = \frac{135}{360} \times \pi \times 169 \]
Next, simplify \( \frac{135}{360} \):
\[ \frac{135}{360} = \frac{135 \div 45}{360 \div 45} = \frac{3}{8} \]
Now, substitute \( \frac{3}{8} \) into the formula:
\[ A = \frac{3}{8} \times \pi \times 169 \]
Now multiply:
\[ A = \frac{3 \times 169}{8} \times \pi = \frac{507}{8} \times \pi \]
Finally, calculate \( \frac{507}{8} \):
\[ \frac{507}{8} = 63.375 \]
Therefore, the area of the sector is:
\[ A \approx 63.375\pi \text{ square inches} \approx 199.1 \text{ square inches (using } \pi \approx 3.14 \text{)} \]
So the area of the sector is approximately:
\[ A \approx 199.1 \text{ square inches} \]
Or you can express it as:
\[ A = \frac{507}{8} \pi \text{ square inches} \]