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 given 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, and \( \theta \) is the angle in radians.

**Step 1: Find the radius.**

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

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

**Step 2: Plug in the values to the area formula.**

We know the area \( A \) of the sector is \( 8\pi \) square feet. Thus, we can set up the equation:

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

Calculating \( 8^2 \):

\[
8^2 = 64
\]

Now substituting that back into the equation:

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

Simplifying:

\[
8\pi = 32\theta
\]

**Step 3: Solve for \( \theta \).**

To find \( \theta \), we rearrange the equation:

\[
\theta = \frac{8\pi}{32}
\]

This simplifies to:

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

**Step 4: Convert radians to degrees.**

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

\[
\theta = \frac{\pi}{4} \cdot \frac{180^\circ}{\pi} = \frac{180^\circ}{4} = 45^\circ
\]

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

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