2.1. 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.
2.1. By the condition, $\angle A C B=90^{\circ}$ (Fig. 2.1); therefore, $B C=\frac{c}{2}$, $A C=\frac{c \sqrt{3}}{2}$, from which $S_{\text {base }}=\frac{1}{2} A C \cdot B C=\frac{c^{2} \sqrt{3}}{8}$. Draw $S O$ such that $A O=O B$. 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 $1^{\circ}$). In $\triangle A S O$ we have $\angle S A O=$
$=45^{\circ}$ and, therefore, $S O=A O=\frac{c}{2}$. Finally, we get

$$
V=\frac{1}{3} S_{\text {base }} \cdot S O=\frac{c^{3} \sqrt{3}}{48}
$$

Answer: $\frac{c^{3} \sqrt{3}}{48}$

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-456.jpg?height=548&width=453&top_left_y=97&top_left_x=704)

Fig. 2.1