To solve the problem, we start by analyzing the regular hexagon (ABCDEF) with side length 1 and the triangle (ACD) formed by its vertices (A), (C), and (D).

Step 1: Assign Coordinates

We place the hexagon centered at the origin with side length 1. Key coordinates are:
(A(1,0)), (C(-0.5, \frac{\sqrt{3}}{2})), (D(-1,0)).

Step 2: Calculate the Area of Triangle (ACD)

The base (AD) is the distance between (A(1,0)) and (D(-1,0)), which is (2).

The height of the triangle is the vertical distance from (C) to the line (AD) (the x-axis), which is the y-coordinate of (C): (\frac{\sqrt{3}}{2}).

Area of triangle (ACD) is:
[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} ]

Step 3: Intersection of Triangle (ACD) and Hexagon

Since the hexagon is convex, the triangle (ACD) (formed by its vertices) lies entirely inside the hexagon. Thus, the intersection area equals the area of (ACD).

Answer: (\boxed{\dfrac{\sqrt{3}}{2}})

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