jmc

Let $A$ be the area of the circular dartboard. If the measure of a central angle of a sector is $x$ degrees, then the area of the sector is $\left(\frac{x}{360}\right)A$. The probability of the dart landing in a region is ratio of the area of the region to the area of the dartboard, so $$\frac{1}{6}=\frac{\left(\frac{x}{360}\right)A}{A}.$$ Solve to find $x=\boxed{60}$.