Solution:
The midpoint of the side of the hexagon on the diameter is the center of the circle. Draw the segment from this center to a vertex of the hexagon on the circle. This segment, whose length is $1/2$, is the hypotenuse of a right triangle whose legs have lengths $a/2$ and $a\sqrt{3}$, where $a$ is a side of the hexagon. So $1/4 = a^{2}(1/4 + 3)$, so $a^{2} = 1/13$. The hexagon consists of $6$ equilateral triangles of side length $a$, so the area of the hexagon is $3a^{2}\sqrt{3}/2 = 3\sqrt{3}/26$.