To determine the length of the arc created by a 30-degree central angle in a circle with a 2-inch diameter, we can use the formula for the arc length:
\[ s = r \theta \]
Where:
- \(s\) is the arc length,
- \(r\) is the radius of the circle,
- \(\theta\) is the central angle in radians.
Step 1: Find the radius
The diameter \(d\) of the circle is given as 2 inches. The radius \(r\) is half of the diameter:
\[ r = \frac{d}{2} = \frac{2 \text{ inches}}{2} = 1 \text{ inch} \]
Step 2: Convert the angle from degrees to radians
We need to convert the angle from degrees to radians. The conversion is done using the fact that \(360\) degrees is equal to \(2\pi\) radians:
\[ \theta = 30^\circ \times \left( \frac{2\pi \text{ radians}}{360^\circ} \right) = 30^\circ \times \left( \frac{\pi}{180} \right) = \frac{30\pi}{180} = \frac{\pi}{6} \text{ radians} \]
Step 3: Calculate the arc length using the formula
Now, substitute \(r\) and \(\theta\) into the arc length formula:
\[ s = r \theta = 1 \text{ inch} \times \frac{\pi}{6} \text{ radians} = \frac{\pi}{6} \text{ inches} \]
Conclusion
The length of the arc created by a 30 degree central angle in a circle with a 2-inch diameter is
\[ s = \frac{\pi}{6} \text{ inches}. \]