Question

The radius of a circle is 3 feet. What is the length of a 135° arc?

Answer 1

To find the length of an arc in a circle, you can use the formula:

\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]

where:
- \(\theta\) is the central angle in degrees,
- \(r\) is the radius of the circle.

In this case, the radius \(r\) is 3 feet and the central angle \(\theta\) is 135°.

Now, plug in the values:

1. Calculate the circumference of the circle:
\[
2\pi r = 2\pi \times 3 = 6\pi \text{ feet}
\]

2. Now calculate the arc length:
\[
\text{Arc Length} = \frac{135}{360} \times 6\pi
\]

3. Simplify the fraction:
\[
\frac{135}{360} = \frac{135 \div 45}{360 \div 45} = \frac{3}{8}
\]

4. Now plug this back into the arc length formula:
\[
\text{Arc Length} = \frac{3}{8} \times 6\pi = \frac{18\pi}{8} = \frac{9\pi}{4} \text{ feet}
\]

Thus, the length of the 135° arc is

\[
\frac{9\pi}{4} \text{ feet} \approx 7.0686 \text{ feet} \text{ (if you need a decimal approximation)}
\]

Final answer:
\[
\text{Length of the arc} = \frac{9\pi}{4} \text{ feet}
\]