To solve the problem of finding the angle measure of an arc with a length of \(3 \pi\) inches in a circle with a radius of 12 inches, follow these steps:
1. Understand the Relationship:
The length of an arc (\(s\)) in a circle is related to the radius (\(r\)) and the angle subtended by the arc at the circle's center (\(\theta\)) using the formula:
[tex]\[
s = r \cdot \theta
\][/tex]
2. Given Values:
- The radius \(r\) is 12 inches.
- The arc length \(s\) is \(3 \pi\) inches.
3. Set up the Equation:
Substitute the given values into the formula:
[tex]\[
3\pi = 12 \cdot \theta
\][/tex]
4. Solve for \(\theta\):
Isolate \(\theta\) on one side of the equation by dividing both sides by 12:
[tex]\[
\theta = \frac{3\pi}{12} = \frac{\pi}{4}
\][/tex]
So, the angle \(\theta\) in radians is \(\frac{\pi}{4}\).
To provide the angle in degrees (for those who prefer degrees):
1. Convert Radians to Degrees:
Use the conversion factor \(180^\circ = \pi\) radians.
[tex]\[
\theta = \frac{\pi}{4} \times \frac{180^\circ}{\pi} = 45^\circ
\][/tex]
Thus, the angle measure of an arc \(3 \pi\) inches long in a circle with a radius of 12 inches is \( \frac{\pi}{4} \) radians or \( 45^\circ \).
- Exact answer in radians: \(\frac{\pi}{4}\)
- Answer in degrees: [tex]\(45^\circ\)[/tex]
