A pyramid's base is a square with unit side length. One of its side edges is also of unit length and coincides with the height of the pyramid. What is the largest dihedral angle?
The base of the pyramid is $ABCD$, its height is $ED$. We derived the pyramid from a unit cube; $AE=EC=\sqrt{2}$, which is the face diagonal of the cube, and $EB=\sqrt{3}$ is the space diagonal of the cube. The planes $AED$ and $CED$ are perpendicular to the base plane and to each other, the planes $AEB$ and $BEC$ form a $45^{\circ}$ angle with the base plane, and an acute angle with the planes $AED$ and $CED$ respectively.

![](https://cdn.mathpix.com/cropped/2024_05_02_7dcb49fa097d9bde47f6g-1.jpg?height=504&width=439&top_left_y=306&top_left_x=846)

Therefore, the angle formed by the planes $AEB$ and $BEC$ is the largest dihedral angle of the pyramid. To calculate it, we draw perpendiculars from $A$ and $C$ to the edge $EB$, which measure the inclination of the two planes. From the symmetry of the cube, it follows that the two perpendiculars intersect the edge $EB$ at the same point $F$.

![](https://cdn.mathpix.com/cropped/2024_05_02_7dcb49fa097d9bde47f6g-1.jpg?height=228&width=409&top_left_y=965&top_left_x=869)

First, we calculate the length of the segment $AF$ in the triangle $AEB$ using the right triangles $AFB$ and $AFE$:

$$
AF^{2}=1-x^{2}
$$

$$
AF^{2}=(\sqrt{2})^{2}-(\sqrt{3}-x)^{2}
$$

From this, $x=\frac{1}{\sqrt{3}}$ and $AF=\sqrt{1-\left(\frac{1}{\sqrt{3}}\right)^{2}}=\sqrt{\frac{2}{3}}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_7dcb49fa097d9bde47f6g-1.jpg?height=208&width=328&top_left_y=1544&top_left_x=904)

We apply the cosine rule to the triangle $AFC$:

$$
\cos \alpha=\frac{\frac{2}{3}+\frac{2}{3}-2}{2 \cdot \sqrt{\frac{2}{3}} \cdot \sqrt{\frac{2}{3}}}=-\frac{1}{2}
$$

From this, $\alpha=120^{\circ}$, which is the largest dihedral angle.

Remark. Three such congruent pyramids can be assembled into a cube. The three congruent dihedral angles around the space diagonal of the cube fill the space without gaps, i.e., each is exactly one third of $360^{\circ}$, or $120^{\circ}$. The other dihedral angles of the pyramid are all smaller than $120^{\circ}$.