22. The diagram shows a design formed by drawing six lines in a regular hexagon. The lines divide each edge of the hexagon into three equal parts.
What fraction of the hexagon is shaded?
A $\frac{1}{5}$
B $\frac{2}{9}$
C $\frac{1}{4}$
D $\frac{3}{10}$
E $\frac{5}{16}$
Solution: $\mathbf{B}$
If we draw in the additional lines shown in the diagram on the right, the hexagon is divided into 54 small congruent equilateral triangles of which 12 are shaded. So the fraction that is shaded is $\frac{12}{54}=\frac{2}{9}$.
[In fact, it is easier to note that the hexagon can be divided into 6 congruent equilateral triangles, like the one shown with the bold edges. Each of these is made up of 9 of the small equilateral triangles of which 2 are shaded.]