To find the area \( A \) of a sector, you can use the formula:
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]
where:
- \( \theta \) is the angle in degrees,
- \( r \) is the radius.
For this problem, the angle \( \theta = 135^\circ \) and the radius \( r = 13 \) in.
Now, substitute the values into the formula:
\[ A = \frac{135}{360} \times \pi \times (13)^2 \]
First, calculate \( (13)^2 \):
\[ (13)^2 = 169 \]
Now plug that into the formula:
\[ A = \frac{135}{360} \times \pi \times 169 \]
Simplifying \( \frac{135}{360} \):
\[ \frac{135}{360} = \frac{135 \div 45}{360 \div 45} = \frac{3}{8} \]
So, we have:
\[ A = \frac{3}{8} \times \pi \times 169 \]
Using \( \pi \approx 3.14159 \):
\[ A \approx \frac{3}{8} \times 3.14159 \times 169 \approx \frac{3 \times 3.14159 \times 169}{8} \]
Calculating the area step-by-step:
-
Calculate \( 3 \times 3.14159 \): \[ 3 \times 3.14159 \approx 9.42477 \]
-
Multiply by \( 169 \): \[ 9.42477 \times 169 \approx 1591.15653 \]
-
Divide by \( 8 \): \[ \frac{1591.15653}{8} \approx 198.89456625 \]
So, rounding this value gives us:
\[ A \approx 198.9 , \text{in}^2 \]
When comparing with the options given, the closest value is:
A = 199.1 in.²
Therefore, the correct response is:
A = 199.1 in.²