# Solution to the Two Hexagons and a Square Puzzle +-- {.image} [[TwoHexagonsandaSquare.png:pic]] > Two regular hexagons, and a square of side length $2$. What's the shaded area? =-- ## Solution by [[Lengths in a Regular Hexagon]] +-- {.image} [[TwoHexagonsandaSquareLabelled.png:pic]] =-- The side length of the square is the same as the smaller diameter of the inner hexagon, which is $\sqrt{3}$ times the side length. So the side length of the inner hexagon is $\frac{2}{\sqrt{3}}$. The excess $C D$ is half of what is left over, so has length: $$ \frac{1}{2} \left(2 - \frac{2}{\sqrt{3}}\right) = 1 - \frac{1}{\sqrt{3}} $$ This is the same as the length of $A B$. Since the length of $O A$ is the same as the side length of the inner hexagon, the length of $O B$, which is the same as the side length of the outer hexagon, is: $$ \frac{2}{\sqrt{3}} + 1 - \frac{1}{\sqrt{3}} = 1 + \frac{1}{\sqrt{3}} $$ The [[area of a hexagon]] is $\frac{3\sqrt{3}}{2}$ times the square of its side length, so the area of the shaded region is: $$ \frac{3\sqrt{3}}{2} \left( \left(1 + \frac{1}{\sqrt{3}}\right)^2 - \left(1 + \frac{1}{\sqrt{3}}\right)^2 \right) = 6 $$