Notes
three regular hexagons solution

Three Regular Hexagons

Solution by Lengths in a Regular Hexagon

Let $a$, $b$, $c$ be the side lengths of the three hexagons in ascending order. Using the relationships between the lengths in a regular hexagon, the height of the middle hexagon is $\sqrt{3}$ times its own side length and is also $c$, so $c = \sqrt{3} b$. The distance across a hexagon from vertex to vertex is twice the side length so the height of the largest hexagon is $2 b + 2 a$, so $\sqrt{3} c = 2 b + 2 a$. This shows that $b = 2 a$, and so $c = 2\sqrt{3} a$.

The area of the middle hexagon is therefore $4$ times that of the smallest, and the area of the largest hexagon is $12$ times that of the smallest. The two side triangles consist of one third of the area of the full hexagon, so are $4$ times the area of the smallest. Therefore the shaded area comprises $9$ times that of the smallest hexagon. The fraction that is shaded is then $\frac{9}{12} = \frac{3}{4}$ of the outer hexagon.