Make sure you are using the formula $s = r\theta$ with $\theta$ in radians, not a degree-based formula like $\dfrac{\theta}{360} \cdot 2\pi r$.

After substituting $s = 12\pi$ and $\theta = \dfrac{2\pi}{3}$ into $s = r\theta$, solve for $r$ by dividing the arc length by the central angle. Be careful with dividing by a fraction.

In Desmos, type the expression 12pi / ((2pi)/3) and look at the numerical result that Desmos gives. That value is the radius of the circle in centimeters.

For a circle, when the central angle $\theta$ is in radians, the arc length $s$ is given by

where $r$ is the radius, $\theta$ is the central angle in radians, and $s$ is the length of the intercepted arc.

From the problem:

• The central angle is $\theta = \dfrac{2\pi}{3}$ radians.
• The intercepted arc length is $s = 12\pi$ centimeters.

Substitute these into the formula $s = r\theta$:

To find $r$, divide both sides of the equation by $\dfrac{2\pi}{3}$:

Rewrite this division as multiplication by the reciprocal:

Now simplify:

• $12 \cdot 3 = 36$
• $36 \div 2 = 18$
• $\pi$ cancels with $\pi$

So $r = 18$.

The radius of the circle is $18$ centimeters.