Question

to the nearest square inch, what is the area of the shaded sector in the circle shown below? image of a circle with radius 10 in and central angle 133° for the shaded sector
options:
a. 314 in²
b. 23 in²
c. 116 in²
d. 464 in²

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( A = \frac{\theta}{360^\circ} \times \pi r^2 \).

Step2: Identify the values of \( r \) and \( \theta \)

From the problem, the radius \( r = 10 \, \text{in} \) and the central angle \( \theta = 133^\circ \).

Step3: Substitute the values into the formula

First, calculate \( \pi r^2 \): \( \pi \times (10)^2 = 100\pi \).
Then, calculate the fraction of the circle the sector represents: \( \frac{133^\circ}{360^\circ} \).
Multiply these two results: \( A = \frac{133}{360} \times 100\pi \).

Step4: Compute the numerical value

Using \( \pi \approx 3.1416 \), we have:
\( A \approx \frac{133}{360} \times 100 \times 3.1416 \)
\( \approx \frac{133 \times 314.16}{360} \)
\( \approx \frac{41783.28}{360} \)
\( \approx 116.06 \, \text{in}^2 \), which rounds to \( 116 \, \text{in}^2 \).

Answer:

C. \( 116 \, \text{in}^2 \)