Let $ABCDEFGH$ be a cube of edge length $4$ with bottom face $ABCD$ and top face $EFGH$ ($E$ above $A$, $F$ above $B$, and so on). Let $P$, $Q$, $R$ be the midpoints of $\overline{AB}$, $\overline{CG}$, and $\overline{EH}$. What is the area of $\triangle PQR$?

1. Use $A=(0,0,0)$, $B=(4,0,0)$, $C=(4,4,0)$, $D=(0,4,0)$ and $E,F,G,H$ at height $4$ above them. Then $P=(2,0,0)$, $Q=(4,4,2)$, $R=(0,2,4)$.
2. $PQ^2=2^2+4^2+2^2=24$, $QR^2=4^2+2^2+2^2=24$, $RP^2=2^2+2^2+4^2=24$, so $\triangle PQR$ is equilateral with side $2\sqrt6$.
3. Area $=\frac{\sqrt3}4\cdot24=\boxed{6\sqrt3}$.