Given:
ABCDEF is a regular hexagon.

To find:
The ratio of the area of triangle ACE to that of the hexagon ABCDEF.

Solution:
Step 1: Find the area of the regular hexagon ABCDEF.
The formula to find the area of a regular hexagon is:
Area = (3√3/2) * (Side)^2
Since ABCDEF is a regular hexagon, all its sides are equal.
Let's assume the side length of the hexagon is 'a'. Therefore,
Area of ABCDEF = (3√3/2) * a^2

Step 2: Find the area of triangle ACE.
To find the area of a triangle, we can use the formula:
Area = (1/2) * base * height
In triangle ACE, the base is AC (which is equal to the side length of the hexagon 'a') and the height can be found by drawing a perpendicular from point C to line segment AE. Let's call this point O. Since ABCDEF is a regular hexagon, angle AOC is 120 degrees. Therefore, angle OAC is 60 degrees. Now, triangle AOC is an equilateral triangle because all sides of the hexagon are equal. Hence, angle OCA is also 60 degrees. Therefore, triangle OCA is also an equilateral triangle.
Let's assume the height of triangle ACE is 'h'. In triangle OCA, the height is h/2. Using trigonometry, we can find the value of h/2:
tan 60 = (h/2) / (a/2)
√3 = h / a
h = √3a

Now, we can find the area of triangle ACE:
Area of ACE = (1/2) * a * √3a

Step 3: Find the ratio of the area of triangle ACE to that of the hexagon ABCDEF.
Ratio = (Area of ACE) / (Area of ABCDEF)
Ratio = [(1/2) * a * √3a] / [(3√3/2) * a^2]
Ratio = (√3/2) / (3√3/2)
Ratio = (√3/2) * (2/3√3)
Ratio = 1/3

Therefore, the ratio of the area of triangle ACE to that of the hexagon ABCDEF is 1/3.

Hence, the correct answer is option 'A' - 1/3.