To find the degree measure of the central angle corresponding to an arc length, we can use the formula for the length of an arc:
\[ L = r \theta \]
where:
- \(L\) is the length of the arc,
- \(r\) is the radius of the circle, and
- \(\theta\) is the central angle in radians.
We can rearrange this formula to solve for \(\theta\):
\[ \theta = \frac{L}{r} \]
Given:
- \(L = 44\) inches,
- \(r = 12\) inches.
Substituting the given values:
\[ \theta = \frac{44}{12} = \frac{11}{3} \text{ radians} \]
To convert radians to degrees, we use the conversion factor \( \frac{180^\circ}{\pi} \):
\[ \theta_{degrees} = \theta \times \frac{180^\circ}{\pi} \]
Substituting \(\theta = \frac{11}{3}\):
\[ \theta_{degrees} = \frac{11}{3} \times \frac{180^\circ}{\pi} \]
Calculating this gives:
\[ \theta_{degrees} \approx \frac{11 \times 180}{3 \times 3.14159} \approx \frac{1980}{9.42477} \approx 210.3^\circ \]
Therefore, the degree measure of the central angle, rounded to the nearest tenth of a degree, is \(210.3^\circ\).
4 answers