Notes
three circles in a semi-circle solution

Solution to the Three Circles in a Semi-Circle Puzzle

Solution by Lengths in a Regular Hexagon

By reflecting the diagram to create a full circle, it can be seen that the radius of the outer semi-circle is three times the radius of the smaller circle. This comes from the fact that joining the centres of the small circles makes a hexagon.

Therefore the area of the semi-circle is $\frac{1}{2} \times 9$ times the area of one of the smaller circles. Since there are three such circles in the semi-circle, the area of the outer semi-circle is $\frac{3}{2}$ times the unshaded area, so the area of the shaded region is $\frac{1}{3}$ of the semi-circle.