If d is the diameter of the small circle, then diameter of big circle is 2d.
Perimeter of shaded figure = circumference of big circle + circumference of small circle
Circumference of big circle -> π(2d) = 2πd
Circumference of small circle -> πd
2πd + πd = 30π
3πd = 30π
d = 30π ÷ 3π = 10 cm