Questions: Using the given information, find the area of the sector of a circle of radius r formed by a central angle θ. Radius, r Central Angle, θ 2 inches θ=240° The area of the sector is square inches. (Simplify your answer. Type an exact answer in terms of π.) The area of the sector is approximately square inches. (Round to two decimal places as needed.)

Solution Steps

To find the area of a sector of a circle, we use the formula: \[ \text{Area} = \frac{\theta}{360} \times \pi r^2 \] where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. We will first calculate the exact area in terms of \(\pi\), and then compute the approximate area by evaluating the expression numerically.

We are given:

• Radius \( r = 2 \) inches
• Central angle \( \theta = 240^\circ \)

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

Substitute the given values into the formula: \[ \text{Exact Area} = \frac{240}{360} \times \pi \times 2^2 \] \[ \text{Exact Area} = \frac{2}{3} \times \pi \times 4 \] \[ \text{Exact Area} = \frac{8}{3} \pi \]

Evaluate the exact area numerically: \[ \text{Approximate Area} = \frac{8}{3} \pi \approx 8.3776 \text{ square inches} \]

Final Answer