4B. A cube is inscribed in a right regular pyramid such that four of its edges are on the lateral faces of the pyramid, and the other four edges are on the base of the pyramid.

Determine the volume and surface area of the cube if the pyramid has height $h$ and base edge $a$.
Solution. Let's denote the pyramid as $S A B C D$, and the cube inscribed in it as $A_{1} B_{1} C_{1} D_{1} A_{2} B_{2} C_{2} D_{2}$. Let the side length of the cube be $x$. From the similarity of $\triangle S O_{1} B \sim \triangle S O B_{1}$, we get

$$
\overline{S O}: \overline{S O_{1}}=\overline{O B_{1}}: \overline{O_{1} B}
$$

We know that

$$
S O_{1}=h, \overline{S O}=h-x, \overline{O N}=\frac{x}{2}, \overline{O_{1} B}=\frac{a}{\sqrt{2}}
$$

from which we obtain

$$
\frac{h-x}{h}=\frac{x}{a}, x=\frac{a h}{a+h}
$$

From here, the surface area and volume of the cube are given by

$$
P=6 x^{2}=6\left(\frac{a h}{a+h}\right)^{2}, \quad V=x^{3}=\left(\frac{a h}{a+h}\right)^{3}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_56aeac1763a3f1e8ce2ag-03.jpg?height=697&width=552&top_left_y=262&top_left_x=916)

## II year