Circle surrounded by many circles!

the circumference of outer circle is: $2 \pi (r+\frac{r}{2}) = 3 \pi r$ since circles touch each other diistance between points of contact is r = r/2 + r/2; Hence $3 \pi = 9$ such circles can be joined together.

So, if you connect the centers of all possible externally tangent circles with radius .5r, you get a circle with radius 1.5r around the center of the original circle. This circle must have a radius of 3pi*r, or ~9.42r. The externally tangent circles with radius .5r all have diamater r. The big circle with the diameter 1.5r passes through the externally tangent circles slightly below the diameter line, so I guessed that it would be 9, since brilliant only takes whole numbers. Would love to hear a more mathematically sound answer though.