12.343. The linear angle of the dihedral angle formed by two adjacent lateral faces of a regular quadrilateral pyramid is twice the plane angle at the vertex of the pyramid. Find the plane angle at the vertex of the pyramid.
## Solution.

Let $SO$ be the height of the regular pyramid $SABCD$ (Fig. 12.204), $\angle BED$ be the linear angle of the dihedral angle between the adjacent lateral faces $BSC$ and $DSC$. Then $DE \perp SC, EO \perp BD, \angle OED = \frac{1}{2} \angle BED$.

If $F$ is the midpoint of $CD$, then $SF \perp CD, \angle D SF = \frac{1}{2} \angle DSC$.

Let $SD = a, \angle DSC = \alpha$. Then $\angle BED = 2\alpha, \angle OED = \alpha$.

In $\triangle DFS (\angle DFS = 90^\circ): DF = a \sin \frac{\alpha}{2}$, and from $\triangle DES (\angle DES = 90^\circ): DE = a \sin \alpha$.

In $\triangle EOD (\angle EOD = 90^\circ): OD = DE \sin \alpha = a \sin \alpha$.

Since $ABCD$ is a square, then $OD = DF \sqrt{2} \Rightarrow$

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1102.jpg?height=773&width=748&top_left_y=27&top_left_x=173)

Fig. 12.204

$\Rightarrow a \sin^2 \alpha = a \sqrt{2} \sin \frac{\alpha}{2} \Leftrightarrow \sin^2 \alpha = \sqrt{2} \sin \frac{\alpha}{2} \Rightarrow \sin^4 \alpha = 2 \sin^2 \frac{\alpha}{2} \Leftrightarrow$

$\Leftrightarrow (1 - \cos^2 \alpha)(1 - \cos^2 \alpha) = 1 - \cos \alpha \Leftrightarrow$

$\Leftrightarrow (1 - \cos^2 \alpha)(1 + \cos \alpha)(1 - \cos \alpha) = 1 - \cos \alpha$.

Since $\cos \alpha \neq 1$, then $(1 - \cos^2 \alpha)(1 + \cos \alpha) = 1 \Leftrightarrow$

$1 + \cos \alpha - \cos^2 \alpha - \cos^3 \alpha = 1 \Leftrightarrow$

$\Leftrightarrow \cos^3 \alpha + \cos^2 \alpha - \cos \alpha = 0$.

Since $0 < \cos \alpha < 1$, then $\cos^2 \alpha + \cos \alpha - 1 = 0$;

$\cos \alpha = \frac{\sqrt{5} - 1}{2}$
$\alpha = \arccos \frac{\sqrt{5} - 1}{2}$.

Answer: $\arccos \frac{\sqrt{5} - 1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1103.jpg?height=662&width=601&top_left_y=34&top_left_x=93)

Fig. 12.205

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-1103.jpg?height=625&width=525&top_left_y=79&top_left_x=737)

Fig. 12.206