# Nested Circles and Polygons II +-- {.image} [[NestedCirclesandPolygonsII.png:pic]] > Both polygons are regular. What fraction of the design is shaded? =-- ## 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 $\frac{3}{8}$.