Question

the radius of a circle is 8 feet. what is the angle measure of an arc π feet long?
l=π ft
r=8 ft
give the exact answer in simplest form.

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( l \) of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( l=\frac{\theta}{360}\times2\pi r \).

Step2: Substitute known values

We know \( l = \pi \) feet and \( r = 8 \) feet. Substitute these into the formula:
\( \pi=\frac{\theta}{360}\times2\pi\times8 \)

Step3: Solve for \( \theta \)

First, simplify the right - hand side: \( \frac{\theta}{360}\times16\pi=\frac{16\pi\theta}{360} \)
So the equation becomes \( \pi=\frac{16\pi\theta}{360} \)
Divide both sides by \( \pi \) (since \( \pi
eq0 \)): \( 1 = \frac{16\theta}{360} \)
Then cross - multiply: \( 16\theta=360 \)
Solve for \( \theta \): \( \theta=\frac{360}{16}=\frac{45}{2} = 22.5 \)

Answer:

\( \frac{45}{2} \) (or \( 22.5 \))