An arc of length 135 ft subtends a central angle 𝜃 in a circle of radius 45 ft. Find the measure of 𝜃 in degrees. (Round your answer to one decimal place.)

To find the measure of the central angle \(\theta\) in degrees, we can use the formula that relates the arc length \(s\), the radius \(r\), and the angle \(\theta\) in radians:

\[
\theta = \frac{s}{r}
\]

Given:
- Arc length \(s = 135\) ft
- Radius \(r = 45\) ft

Substitute the values into the formula:

\[
\theta = \frac{135}{45} = 3 \text{ radians}
\]

Now, to convert the angle from radians to degrees, use the conversion factor:

\[
1 \text{ radian} = \frac{180}{\pi} \text{ degrees}
\]

So,

\[
\theta \text{ (in degrees)} = 3 \times \frac{180}{\pi}
\]

\[
\theta \approx 3 \times 57.2958 \approx 171.9 \text{ degrees}
\]

Thus, the measure of \(\theta\) is approximately \(171.9\) degrees.

Would you like further details or have any questions?  

Here are five related questions:

1. How would the angle change if the radius were doubled?
2. What would the angle be if the arc length was 90 feet instead?
3. How is the formula for arc length derived?
4. How do you convert degrees back to radians?
5. What is the difference between a central angle and an inscribed angle?

**Tip:** When converting radians to degrees, always remember the conversion factor \( \frac{180}{\pi} \).

Learn how to calculate the central angle \( \theta \) of a circle given the arc length \( s = 135 \) ft and radius \( r = 45 \) ft. Step-by-step solution using the central angle formula and radian to degree conversion. Suitable for Grades 10-12.

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