Questions: Find the area of the sector of a circle with radius 3 miles formed by a central angle of 230 degrees: square miles Round your answer to two decimal places.

Solution Steps

To find the area of a sector of a circle, we can use the formula: \[ \text{Area} = \frac{\theta}{360} \times \pi \times r^2 \] where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle. Given \(\theta = 230^\circ\) and \(r = 3\) miles, we can plug these values into the formula and calculate the area.

We are given the radius \( r = 3 \) miles and the central angle \( \theta = 230^\circ \).

The formula for the area of a sector is: \[ \text{Area} = \frac{\theta}{360} \times \pi \times r^2 \]

Substituting \( \theta = 230 \) and \( r = 3 \) into the formula, we get: \[ \text{Area} = \frac{230}{360} \times \pi \times 3^2 \]

First, calculate the fraction: \[ \frac{230}{360} \approx 0.6389 \]

Next, calculate \( \pi \times 3^2 \): \[ \pi \times 9 \approx 28.2743 \]

Then, multiply the results: \[ 0.6389 \times 28.2743 \approx 18.0642 \]

Rounding \( 18.0642 \) to two decimal places, we get: \[ 18.06 \]

Final Answer