In the figure below, $ABCD$ is a parallelogram, $AD=18$ and $DC=27$. Segments $\overline{EG}$ and $\overline{FH}$ are perpendicular to sides of the parallelogram as shown, and $EG=16$. What is the area of the parallelogram? [asy]
size(7cm);
pair b=(-sqrt(68),16); pair g=(0,0); pair s=(27,0); pair y=b+s;
pair a=(8,16); pair u=(8,0);
pair o=g+0.15*(b-g);
pair e=o+(64/3,4/3*sqrt(68));
dot(b); dot(o); dot(g); dot(u); dot(s); dot(a); dot(y); dot(e);
draw(y--e);
draw(g--o);
draw(o--e, dashed);
draw(a--u, dashed);
draw(o--b, MidArrow(TeXHead));
draw(s--e, MidArrow(TeXHead));
draw(b--b+0.94*(y-b),MidArrow(TeXHe ad));
draw(y+0.98*(b-y)--y, MidArrow(TeXHead));
draw(g--g+0.96*(s-g),MidArrow(TeXHe ad));
draw(s+0.96*(g-s)--s, MidArrow(TeXHead));
draw(rightanglemark(a, u,g,50));
draw(rightanglemark(o, e,s,50));
label("$A$",b, NW);
label("$D$",g, SW);
label("$C$",s, SE);
label("$B$",y, NE);
label("$H$",o, WSW);
label("$F$",e, ENE);
label("$E$",a, N);
label("$G$",u, S);
[/asy]