Details

Joining the center of the orange disk O to the centers P and Q of two successive purple disks shows that the center angle is , where is the number of purple disks. The line segment connecting O to the tangent point of two consecutive purple disks bisects this angle, so each smaller angle measures . The line segment connecting P and Q is perpendicular to the line segment between the center of the orange disk and the tangent point of the two blue balls. Therefore:

.

Simplifying, .