Call the length of one side of the hexagon $s$ and the height of the hexagon $h$:

So the area of the shaded triangle is $$\frac{1}{2} \times s \times h$$

Now divide the hexagon into $6$ equilateral triangles:

Each triangle has area $$\frac{1}{2}\times s \times \frac{h}{2}$$ so the area of the hexagon is $$6 \times \frac{1}{2}\times s \times \frac{h}{2}=3 \times \frac{1}{2} \times s \times h$$ or $$3 \times \text{Shaded area}$$

So the shaded area is $\frac{1}{3}$ of the area of the whole hexagon.

*This problem is taken from the UKMT Mathematical Challenges.*