AMC 12 2008b Problem 4

On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. What is the ratio of the area of the smaller sector $COD$ to the area of the circle?

unitsize(6mm);
defaultpen(linewidth(0.7)+fontsize(8pt));

pair C = 3*dir (30);
pair D = 3*dir (135);
pair A = 3*dir (0);
pair B = 3*dir(180);
pair O = (0,0);
draw (Circle ((0, 0), 3));
label ("$C$", C, NE);
label ("$D$", D, NW);
label ("$B$", B, W);
label ("$A$", A, E);
label ("$O$", O, S);
label ("$45^\circ$", (-0.3,0.1), WNW);
label ("$30^\circ$", (0.5,0.1), ENE);
draw (A--B);
draw (O--D);
draw (O--C);

(Solution)

\textbf{(A)}\ \frac {2}{9} \textbf{(B)}\ \frac {1}{4} \textbf{(C)}\ \frac {5}{18} \textbf{(D)}\ \frac {7}{24} \textbf{(E)}\ \frac {3}{10}