Once you use the inscribed-angle relationship, what is the degree measure of the minor arc $PR$ that corresponds to a $40^\circ$ inscribed angle?

Arc length is a fraction of the total circumference. That fraction is $\dfrac{\text{arc measure in degrees}}{360}$. How can you apply this using the arc measure you found and the circumference $36\pi$?

Write an expression of the form $\dfrac{\text{arc measure}}{360} \cdot 36\pi$ and then simplify it carefully to match one of the answer choices.

In Desmos, type 2*40 to verify that the intercepted arc measure, in degrees, is twice the $40^\circ$ inscribed angle. Note the degree measure you get.

In a new line, type (2*40/360)*36*pi (or equivalently (80/360)*36*pi if you already know the arc is 80°). Look at the simplified result that Desmos shows and then match that value to one of the answer choices.

Angle $\angle PQR$ is an inscribed angle, which means its vertex is on the circle and its sides (rays) go through points $P$ and $R$.

Key fact: the measure of an inscribed angle is half the measure of its intercepted arc.

So if $m\angle PQR = 40^\circ$, then the measure of arc $PR$ is

This $80^\circ$ is the central-angle measure (in degrees) of the minor arc $PR$.

The total circle is $360^\circ$, and its circumference is given as $36\pi$ centimeters.

Arc length is the same fraction of the circumference as the arc’s degree measure is of $360^\circ$:

Substitute the values you know:

Now simplify

First reduce the fraction $\tfrac{80}{360}$:

So the arc length is

Then

Therefore, the length of the minor arc $PR$ is $8\pi$ centimeters.