Let $ABCD$ be a regular tetrahedron. Let $E$, $F$, $G$, $H$ be the centers of faces $BCD$, $ACD$, $ABD$, $ABC$, respectively. The volume of pyramid $ABCD$ is $18.$ Find the volume of pyramid $DEFG$.
Each center of the face of the side is a centroid. Say I is the midpoint of of side AC. We can form equilateral triangle FIH. We see that FH = FI = 1/3 of the centroid. Thus each length of the smaller tetrahedron must be 1/3 that of the larger tetrahedron. \( \left (1 \over 3 \right )^3 * 18 = \left (2 \over 3 \right )\)