11.001. The base of the pyramid is a right-angled triangle with a hypotenuse equal to \( c \) and an acute angle of \( 30^{\circ} \). The lateral edges of the pyramid are inclined to the base plane at an angle of \( 45^{\circ} \). Find the volume of the pyramid.
## Solution.

By the condition $\angle A C B=90^{\circ}, \angle B A C=30^{\circ}$ (Fig. 11.3); therefore, $B C=\frac{c}{2}, A C=\frac{c \sqrt{3}}{2}$, from which $S_{\text {base }}=\frac{1}{2} A C \cdot B C=\frac{c \sqrt{3}}{8}$. Draw $S O$ such that $A \dot{O}=O B$. Then $O$ is the center of the circumscribed circle around $\triangle A B C$, and $S O$ is the height of the pyramid (see "Additional relationships between elements of prisms and pyramids", point 1). In $\triangle A S O$ we have $\angle S A O=45^{\circ}$, and thus $S O=A O=\frac{c}{2}$. Therefore,

$$
\begin{aligned}
& V=\frac{1}{3} S_{\text{base}} \cdot S O=\frac{c \sqrt[3]{3}}{48} \\
& \text { Answer: } \quad \frac{c \sqrt[3]{3}}{48}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-650.jpg?height=498&width=418&top_left_y=63&top_left_x=175)

Fig. 11.3

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-650.jpg?height=466&width=422&top_left_y=99&top_left_x=745)

Fig. 11.4