Three Triangles in a Hexagon

What fraction is shaded? The hexagon is regular, with equally spaced dots around its perimeter.

Split the hexagon into triangles as shown on the figure below. Note that each triangle has a side that is half of the side of the hexagon, and a height that is h, 2h, 3h, or 4h, where the distance between two opposite sides of the hexagon is equal to 4h.

Then, if the areas of the shaded triangles are S, 3S, and 4S, the total area of the hexagon is 24S. Thus, the answer is 8S/24S=1/3.