Question

the diameter of a circle is 8 meters. what is the angle measure of an arc bounding a sector with area 2π square meters? give the exact answer in simplest form.

Explanation:

Step1: Calculate radius

Radius \( r = \frac{\text{diameter}}{2} = \frac{8}{2} = 4 \, \text{m} \)

Step2: Sector area formula

Sector area \( A = \frac{\theta}{360^\circ} \cdot \pi r^2 \), given \( A = 2\pi \)

Step3: Solve for θ

\( 2\pi = \frac{\theta}{360^\circ} \cdot \pi (4)^2 \)
Simplify: \( 2 = \frac{16\theta}{360^\circ} \)
\( \theta = \frac{2 \times 360^\circ}{16} = 45^\circ \)

Answer:

45