<p>In the accompanying figure, the outer square <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> has side length <img alt="$40$" height="13" src="https://latex.artofproblemsolving.com/7/8/7/78792fe4680a164f537ea72a4f900e4471433c3b.png" width="17" />. A second square <img alt="$S'$" height="14" src="https://latex.artofproblemsolving.com/b/f/3/bf344aa376979a8a81e8784f92fcb2381897ba65.png" width="15" /> of side length <img alt="$15$" height="13" src="https://latex.artofproblemsolving.com/f/1/9/f1965fae079a9ba2c0726c307070c2355dfcb213.png" width="17" /> is constructed inside <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> with the same center as <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> and with sides parallel to those of <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />. From each midpoint of a side of <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />, segments are drawn to the two closest vertices of <img alt="$S'$" height="14" src="https://latex.artofproblemsolving.com/b/f/3/bf344aa376979a8a81e8784f92fcb2381897ba65.png" width="15" />. The result is a four-pointed starlike figure inscribed in <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" />. The star figure is cut out and then folded to form a pyramid with base <img alt="$S'$" height="14" src="https://latex.artofproblemsolving.com/b/f/3/bf344aa376979a8a81e8784f92fcb2381897ba65.png" width="15" />. Find the volume of this pyramid.</p> <p>                                                        <img alt="" src="/media/question/files/2024/12/21/screenshot-2024-12-21-010749.png" style="width: 268px; height: 258px;" /></p>

Question No: 242

In the accompanying figure, the outer square $S$ has side length $40$ . A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$ . From each midpoint of a side of $S$ , segments are drawn to the two closest vertices of $S'$ . The result is a four-pointed starlike figure inscribed in $S$ . The star figure is cut out and then folded to form a pyramid with base $S'$ . Find the volume of this pyramid.

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