Question

The diameter of a circle is 16 feet. What is the angle measure of an arc bounding a sector with area 8​𝜋 square feet?
K=8​𝜋 sq. ft
d=16 ft
Give the exact answer in simplest form.
°

Answer 1

To find the angle measure of an arc bounding a sector with a known area in a circle, we can use the formula for the area of a sector:

\[
A = \frac{1}{2} r^2 \theta
\]

where:
- \( A \) is the area of the sector,
- \( r \) is the radius of the circle,
- \( \theta \) is the angle in radians.

1. **Determine the radius of the circle**:

The diameter of the circle is given as 16 feet, so the radius \( r \) is:

\[
r = \frac{d}{2} = \frac{16}{2} = 8 \text{ feet}
\]

2. **Set the area of the sector**:

The area of the sector \( A \) is given as \( 8\pi \) square feet.

3. **Substituting into the area formula**:

We set up the equation:

\[
8\pi = \frac{1}{2} \cdot (8)^2 \cdot \theta
\]

Calculating \( (8)^2 \):

\[
(8)^2 = 64
\]

Substituting back into the equation gives:

\[
8\pi = \frac{1}{2} \cdot 64 \cdot \theta
\]

This simplifies to:

\[
8\pi = 32\theta
\]

4. **Solving for \( \theta \)**:

To find \( \theta \):

\[
\theta = \frac{8\pi}{32} = \frac{\pi}{4} \text{ radians}
\]

5. **Convert radians to degrees**:

To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):

\[
\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi}
\]

Substituting \( \theta = \frac{\pi}{4} \):

\[
\theta_{degrees} = \frac{\pi}{4} \times \frac{180}{\pi} = \frac{180}{4} = 45^\circ
\]

Thus, the angle measure of the arc bounding the sector is:

\[
\boxed{45^\circ}
\]