AMC 12 2006b Problem 15

Circles with centers $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ are on the circle centered at $O$, and points $C$ and $D$ are on the circle centered at $P$, such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of hexagon $AOBCPD$?

unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
pair A, B, C, D;
pair[] O;
O[1] = (6,0);
O[2] = (12,0);
A = (32/6,8*sqrt(2)/6);
B = (32/6,-8*sqrt(2)/6);
C = 2*B;
D = 2*A;
draw(Circle(O[1],2));
draw(Circle(O[2],4));
draw((0.7*A)--(1.2*D));
draw((0.7*B)--(1.2*C));
draw(O[1]--O[2]);
draw(A--O[1]);
draw(B--O[1]);
draw(C--O[2]);
draw(D--O[2]);
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SW);
label("$D$", D, NW);
dot("$O$", O[1], SE);
dot("$P$", O[2], SE);
label("$2$", (A + O[1])/2, E);
label("$4$", (D + O[2])/2, E);

$\textbf{(A) } 18\sqrt {3} \textbf{(B) } 24\sqrt {2} \textbf{(C) } 36 \textbf{(D) } 24\sqrt {3} \textbf{(E) } 32\sqrt {2}$