$120^{\circ}$ is what fraction of $360^{\circ}$? Multiply that fraction by the full circumference $2\pi r$ and set it equal to the given arc length $6\pi$.

After you set up the equation with $6\pi$ on one side and an expression involving $r$ on the other, isolate $r$ by using inverse operations (like clearing fractions and dividing).

In Desmos, type 6pi / ((120/360)*2pi) as an expression. The numeric value that Desmos outputs for this expression is the radius of the circle in centimeters.

For a circle with radius $r$ and central angle $\theta$ (in degrees), the length of the arc $s$ is

Here, $s = 6\pi$ and $\theta = 120^{\circ}$.

Plug $s = 6\pi$ and $\theta = 120$ into the formula:

Simplify the fraction $\frac{120}{360}$:

so the equation becomes

First simplify the right side:

So we have

Multiply both sides by 3 to clear the denominator:

Divide both sides by $2\pi$:

So, the radius of circle $P$ is $9$ centimeters.