Notes
three circles in a hexagon in a circle solution

Three Circles in a Hexagon in a Circle

Solution by Lengths in a Regular Hexagon

In the above diagram, $E D$ is a diameter of the outer circle and is the length across a regular hexagon and $A C$ is two diameters of the inner circle and is the short diameter of the hexagon. Therefore the length of $A C$ is $\frac{\sqrt{3}}{2}$ times the length of $E D$. So the radius of the smaller circles is $\frac{\sqrt{3}}{4}$ times the radius of the outer circle. The area of one smaller circle is therefore $\frac{3}{16}$ths of the area of the larger and so the fraction that is shaded is $\frac{9}{16}$ths of the circle.