Pointless Circles

The plane is made of small equilateral triangles with points at their vertices.

Within each triangle those circles with a center within distance 1 of a vertex, that is within the orange pie slice, will contain a point. Those with a center within the yellow region will not.

So the problem is to find the percentage of the area that is yellow.

To do that take the area of the triangle, $\frac{\sqrt3}4$ , and subtract from it the area of the tree pie slices which is equal to half a circle with radius $\frac12$ , that is $\frac{\pi}8$ .

Divide the result by area of triangle $\frac{\sqrt3}4$ and you get $\frac{2\sqrt3-\pi}{2\sqrt3}=0.0931=\boxed{9.31}$ percent

After connecting the lattice grid, the circle's center must be in an equilateral triangle. Since the circle has radius $\frac{1}{2}$ , this center must be more than $\frac{1}{2}$ away from any of the vertices of the triangle. The region in which this is true can be found by constructing three circles of radius $\frac{1}{2}$ around the three vertices:

The area of the equilateral triangle is $\frac{\sqrt{3}}{4}$ . The area of each sector in the triangle is $\frac{1}{2}(\frac{\pi}{3})(\frac{1}{2})^2=\frac{\pi}{24}$ . Therefore, the probability is $\frac{\frac{\sqrt{3}}{4}-3(\frac{\pi}{24})}{\frac{\sqrt{3}}{4}} = 1-\frac{\pi}{2\sqrt{3}} \approx \boxed{9.31\%}$