Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?
$\textbf{(A) }\frac{11}{32} \quad\textbf{(B) }\frac{3}{8} \quad\textbf{(C) }\frac{13}{32} \quad\textbf{(D) }\frac{7}{16}\quad \textbf{(E) }\frac{15}{32}$
Solution 1
Since octagon $ABCDEFGH$ is a regular octagon, it is split into $8$ equal parts, such as triangles $\bigtriangleup ABO, \bigtriangleup BCO, \bigtriangleup CDO$, etc.  These parts, since they are all equal, are $\frac{1}{8}$ of the octagon each.  The shaded region consists of $3$ of these equal parts plus half of another, so the fraction of the octagon that is shaded is $\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{16}=\boxed{\textbf{(D) }\dfrac{7}{16}}.$

Solution 2

The octagon has been divided up into $16$ identical triangles (and thus they each have equal area). Since the shaded region occupies $7$ out of the $16$ total triangles, the answer is $\boxed{\textbf{(D)}~\dfrac{7}{16}}$.
-Flare

Solution 3
For starters, what I find helpful is to divide the whole octagon up into triangles as shown here:

Now, it is just a matter of counting the larger triangles. Remember that $\triangle BOX$ and $\triangle XOA$ are not full triangles and are only half for these purposes. We count it up and we get a total of $\frac{3.5}{8}$ of the shape shaded. We then simplify it to get our answer of $\boxed{\textbf{(D)}~\frac{7}{16}}$.