Question

The diameter of a circle is 20 kilometers. What is the angle measure of an arc bounding a sector with area 5​𝜋 square kilometers?
K=5​𝜋 sq. km
d=20 km
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,
- \( \theta \) is the angle in radians.

1. **Calculate the radius:**
The diameter \( d \) of the circle is 20 km, therefore the radius \( r \) is:

\[
r = \frac{d}{2} = \frac{20}{2} = 10 \text{ km}
\]

2. **Set up the equation for the area of the sector:**
We know that the area \( A \) is \( 5\pi \) square kilometers. Substitute the values of \( A \) and \( r \) into the formula:

\[
5\pi = \frac{1}{2} (10^2) \theta
\]

3. **Solve for \( \theta \):**
Simplifying the formula:

\[
5\pi = \frac{1}{2} (100) \theta
\]

\[
5\pi = 50 \theta
\]

To solve for \( \theta \), divide both sides by 50:

\[
\theta = \frac{5\pi}{50} = \frac{\pi}{10} \text{ radians}
\]

4. **Convert radians to degrees:**
To convert from radians to degrees, use the conversion factor \( \frac{180}{\pi} \):

\[
\theta \text{ (in degrees)} = \frac{\pi}{10} \cdot \frac{180}{\pi} = \frac{180}{10} = 18^\circ
\]

Thus, the angle measure of the arc bounding the sector is \( \boxed{18^\circ} \).