163. Find the volume of a truncated square pyramid with a height of 10, and the sides of the upper and lower bases being 2 and 4, respectively.

## Al-Karaji's Problems.
163. Let the section of the pyramid be a trapezoid $ABCD$, where $DC=2$, $AB=4$, $EF=10$ (Fig. 46). We complete the trapezoid to form triangle $DSC$. The height $SE=10$ (from the similarity of triangles $ABS$ and $DSC$). The volume of the truncated pyramid can be found as the difference between the volume of the complete pyramid and the volume of the upper part:

$$
V=\frac{FS}{3} \cdot AB^{2}-\frac{SE}{3} \cdot DC^{2}=\frac{20}{3} \cdot 16-\frac{10}{3} \cdot 4=\frac{280}{3}=93 \frac{1}{3}
$$

The same result, obviously, can be obtained using the standard formula:

$$
V=\frac{H}{3}(B+b+\sqrt{B b})=\frac{10}{3}(16+4+8)
$$

Al-Karhi (11th century), the author of two treatises "Kafi fil Hisab" ("All Known in Arithmetic") and "Al-Fakri" - an extensive work on algebra. In the first treatise, much attention is given to geometry.