Because the plane is parallel to the base, the two pyramids are similar. If the volume scale factor is known, take its cube root to get the corresponding length scale factor.

Use $V=\frac{1}{3}Bh$ with the original pyramid's volume and height. Then apply the square of the length scale factor to that base area.

Algebra is the fastest method because the key idea is the similarity relationship between volume, length, and area. Desmos can quickly verify the arithmetic after the scale factors are identified.

Enter 10368/(24/3) to evaluate the original base area from $V=\frac{1}{3}Bh$.

Enter ((8/27)^(1/3))^2*(10368/(24/3)). This applies the cube root of the volume ratio to get the length scale factor, squares it for the area scale factor, and multiplies by the original base area.

Let $V_s$ be the volume of the smaller pyramid. The frustum has volume $\frac{19}{8}V_s$, so the original pyramid's volume is $V_s+\frac{19}{8}V_s=\frac{27}{8}V_s$. Therefore, $\frac{V_s}{V_{\text{original}}}=\frac{8}{27}$.

The smaller pyramid and the original pyramid are similar because the cutting plane is parallel to the base. A volume scale factor of $\frac{8}{27}$ corresponds to a length scale factor of $\sqrt[3]{\frac{8}{27}}=\frac{2}{3}$. Thus, the corresponding area scale factor is $\left(\frac{2}{3}\right)^2=\frac{4}{9}$.

For a pyramid, $V=\frac{1}{3}Bh$, where $B$ is the base area. Substitute the given volume and height: $10{,}368=\frac{1}{3}B(24)=8B$. Thus, the original base area is $1{,}296$ square inches.

The cross section is the base of the smaller, similar pyramid. Its area is $\frac{4}{9}(1{,}296)=576$ square inches.