Notes
nested circles and polygons ii solution

Nested Circles and Polygons II

Solution by Lengths in a Regular Hexagon and Square

Let the hexagon have side length $x$. Using the relationships between the lengths in a regular hexagon, the height of the hexagon, which is the diameter of the inner circle, is $\sqrt{3} x$ and the side length of the square is $2x$. The diameter of the square is then $2 \sqrt{2} x$, and this is the diameter of the outer circle. The area of the outer circle is therefore $8 \pi x^2$ and of the inner circle is $3 \pi x^2$. The fraction that is shaded is then $\frac{3}{8}$.