11.201 The lateral edges of a triangular pyramid are of equal length and equal to $a$. Of the three planar angles formed by these edges at the apex of the pyramid, two contain $45^{\circ}$, and the third contains $60^{\circ}$. Determine the volume of the pyramid.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-260.jpg?height=544&width=1055&top_left_y=950&top_left_x=89)

Let $FABC$ be the given pyramid with base $ABC$, $\angle AFC = 60^\circ$, $\angle BFC = \angle AFB = 45^\circ$. Drop perpendiculars from $A$ and $C$ to the edge $FB$. They intersect at a single point $K$ due to the equality of $\triangle AFB$ and $\triangle BFC$.

By the given condition, $\triangle AFC$ is equilateral. $\triangle FCK$ and $\triangle FAK$ are isosceles right triangles $\Rightarrow$

$\Rightarrow AC = a, KC = KA = \frac{a}{\sqrt{2}} \Rightarrow \angle AKC = 90^\circ \Rightarrow$ $\Rightarrow S_{AKC} = \frac{a^2}{4}$. By Theorem 11, $AKC \perp FB \Rightarrow V_{FABC} = \frac{1}{3} S_{AKC} \cdot FB = \frac{a^3}{12}$.

Answer: $\frac{a^3}{12}$.