Notes
two circles inside two polygons inside a circle solution

Solution to the Two Circles Inside Two Polygons Inside a Circle Puzzle

Solution by Lengths in Equilateral Triangles, Properties of Chords, and Intersecting Chords Theorem

As the line segment $E F$ is a chord, its perpendicular bisector $A D$ passes through the centre of the circle and so is a diameter.

Let $x$ be the length of half the side of the square, so $x$ is the length of $E C$. Using lengths in an equilateral triangle, the length of $A B$ is $\sqrt{3} x$ so the length of $A C$ is $2 x + \sqrt{3} x$. Write $y$ for the length of $C D$, then using the intersecting chords theorem applied to $E F$ and $A D$, the $x$ and $y$ satisfy:

which rearranges to

The diameter of the outer circle is therefore:

Its area is therefore $4 \pi x^2$.

The radius of the circle in the square is $x$, and the radius of the circle in the equilateral triangle is a third of the length of $A B$, so is $\frac{1}{\sqrt{3}} x$. The area of the shaded regions is therefore $\pi x^2 + \frac{1}{3} \pi x^2 = \frac{4}{3} \pi x^2$.

Therefore $\frac{1}{3}$rd of the outer circle is shaded.