Question

A central angle of a sector in a circle has a radian measure of
π
3
radians and its arc length is
8
3
π
units. What is the measure of the radius of the circle?

units

Answer 1

To find the radius of the circle given the central angle in radians and the arc length, we can use the formula for arc length:

\[
L = r \theta
\]

where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.

From the problem, we have:
- \( L = \frac{8}{3} \pi \) units
- \( \theta = \frac{\pi}{3} \) radians

Substituting the values into the arc length formula gives us:

\[
\frac{8}{3} \pi = r \left( \frac{\pi}{3} \right)
\]

To solve for \( r \), first eliminate \( \pi \) from both sides:

\[
\frac{8}{3} = r \left( \frac{1}{3} \right)
\]

Next, multiply both sides by 3 to isolate \( r \):

\[
8 = r
\]

Therefore, the radius of the circle is:

\[
\boxed{8} \text{ units}
\]