4. The lengths of all edges of a regular quadrilateral pyramid $S A B C D$ (with apex $S$) are equal to $a$. A plane $\Sigma$ is placed through point $A$, which is normal to the plane $A C S$ and passes through the midpoint $M$ of edge $S C$. Find the volume of the quadrilateral pyramid cut off from the given pyramid by the plane $\Sigma$.
Solution. According to the problem, we need to find the volume $V$ of the pyramid SAKMN (see the diagram). The triangular pyramids KSAM and NSAM have equal volumes. If we take $S A M$ as the base of these pyramids, then they have equal heights, $K R$ and $N R$. Therefore,

$$
V=\frac{2}{3} P_{A S M} \cdot \overline{K R}
$$

Since $\measuredangle A S M=90^{\circ}$, it follows that

$$
P_{A S M}=\frac{1}{2} \overline{A S} \cdot \overline{M S}=\frac{a^{2}}{4}
$$

The point $R$ is the centroid of triangle $A C S$, so we have

$$
\overline{S R}: \overline{S O}=2: 3
$$

From the similarity of triangles $S R K$ and $S O B$, we get:

$$
\overline{R K}=\frac{\overline{S R}}{\overline{S O}} \overline{O B}=\frac{2}{3} \cdot \frac{a \sqrt{2}}{2}=\frac{a \sqrt{2}}{3}
$$

Finally, the desired volume will be:

$$
V=\frac{2}{3} P_{A S M} \cdot \overline{R K}=\frac{a^{3} \sqrt{2}}{18}
$$

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