What is the measure of the central angle of a circle with radius 18 centimeters that intercepts a 12π centimeters arc?

If the radius is 18 cm, then the exact circumference of the full circle is 36pi centimeters through the formula
C = 2*pi*r
If you measured the distance around the circle, you'd get 36pi centimeters.

However, we are told that this arc (piece of the circle) is 12pi cm in length if you were to lay it straight or apply a curved ruler to the arc. The question is: what fraction of 36pi is 12pi? A simpler question is: what is the ratio of 12 to 36? Note how I dropped the "pi" terms to make things easier.

Take the ratio of 12 to 36 and we get 12/36 = 1/3
So the full distance around the circle is three times that of the arc length
Flip things around: the arc length is 1/3 that of the circumference.

So this means that the central angle is 1/3 of 360 degrees as there are 360 degrees in a full circle.

Multiply 1/3 by 360 to get
(1/3)*360 = 360/3 = 120

The central angle is 120 degrees
note: 120 degrees is equivalent to 2pi/3 radians

The answer to your question is 1/9