12.451 In a regular quadrilateral pyramid, a plane is drawn through two lateral edges that do not belong to the same face. The ratio of the area of the section to the lateral surface area of the pyramid is $k$. Find the angle between two adjacent lateral faces and the permissible values of $\boldsymbol{k}$.

In a regular quadrilateral pyramid, a plane is drawn through two lateral edges that do not belong to the same face. The ratio of the area of the section to the lateral surface area of the pyramid is $k$. Find the angle between two adjacent lateral faces and the permissible values of $\boldsymbol{k}$.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-319.jpg?height=332&width=336&top_left_y=833&top_left_x=201)

Let $O$ be the center of the base of the given regular pyramid $FABCD$; $FAC$ is the section of the pyramid by a plane passing through $FA$ and $FC$. Drop perpendiculars from vertices $A$ and $C$ to the edge $FD$: they will intersect at one point $K$ (due to the equality of the lateral faces). Let $\angle AKC = x$. By the problem's condition,

$$
S_{FAC} = S_{FBD} = 4k \cdot S_{FCD} \Rightarrow
$$

$$
\Rightarrow OK \cdot FD = 4k \frac{FD \cdot CK}{2} \Rightarrow 2k = \frac{OK}{CK} = \cos \frac{x}{2} \Rightarrow \cos x = 8k^2 - 1
$$

Note that $OK < 45^\circ \Rightarrow x > 90^\circ \Rightarrow 8k^2 - 1 < 0 \Leftrightarrow k < \frac{\sqrt{2}}{4}$.

Answer: $\arccos(8k^2 - 1), 0 < k < \frac{\sqrt{2}}{4}$.