The Ugly Wallpaper

You find an infinite wall that someone with very poor taste has covered with the following wallpaper:

You find it so offensive that you throw a dart at it.

If the probability that the dart lands on any one of the stars is $\frac{a}{b}$ , where $a$ and $b$ are co-prime numbers, find $a + b$ .

Assume that the dart is thrown randomly, the wallpaper tiles infinitely, that all the angles in the diagram are $30°$ , $60°$ , or $120°$ , and that the lines between the stars and triangles have no width (in other words, the dart either lands on a star or triangle).