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 $DEFG$ is $18.$ Find the volume of pyramid $EFGH$

tomtom Dec 26, 2023

#1**0 **

Note that $DEFG$ is similar to $ABCD$ with ratio $\frac12$.

The ratio of their volumes is $(\frac12)^3=\frac18$, so the volume of $ABCD$ is $8(18)=144$

Using the fact that $ABCD$ is a regular tetrahedron with side length $2$, we find that $[ABC]=12\sqrt{2}$, and so

\[[EFGH]=[ABCD]-4[ABC]=144-4\cdot12\sqrt{2}=\boxed{144-48\sqrt{2}}.\]

BuiIderBoi Dec 26, 2023