Given a regular hexagon, its smaller diagonal is equal to a. Find the side of the hexagon and its large diagonal.

We drive the variable x and denote the side of this regular hexagon this way.
Consider a triangle formed by two sides and a smaller diagonal. We write in this triangle the cosine theorem:
a² = x² + x² – 2 * x * x * cos 120 ° = 2x² + x² = 3x² →
x = √ (a² / 3) = a / √3 is the side of a regular hexagon.
One of the properties of a regular hexagon is that the side is equal to the radius of the circumscribed circle.
The large diagonal is the diameter of the circumscribed circle and is equal to two radii or two sides.
In our case: 2a / √3.
Answer: the side is equal to a / √3, the large diagonal is 2a / √3.