AMC 12 2003a Problem 14

Points $K, L, M,$ and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$.

unitsize(2cm);
defaultpen(fontsize(8)+linewidth(0.8));
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);
draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NW);
label("$D$",D,NE);
label("$K$",K,NNW);
label("$L$",L,E);
label("$M$",M,S);
label("$N$",N,W);

$\textrm{(A)}\ 32 \textrm{(B)}\ 16+16\sqrt{3} \textrm{(C)}\ 48 \textrm{(D)}\ 32+16\sqrt{3} \textrm{(E)}\ 64$