Find the volume of an oblique triangular prism, the base of which is an equilateral triangle with side $a$, if the lateral edge of the prism is equal to the side of the base and is inclined to the base plane at an angle of $60^{\circ}$.
Let $K$ be the orthogonal projection of vertex $A$ of the oblique prism $A B C A 1 B 1 C 1$ onto the base plane $A 1 B 1 C 1$, and $A B=B C=A C=A A 1=B B 1=D D 1=a$. According to the problem, $\angle A A 1 K=60^{\circ}$. From the right triangle $AKA1$, we find that

$$
A K=A A 1 \sin \angle A A 1 K=a \sin 60^{\circ}=\frac{a \sqrt{3}}{2}
$$

Since $A K$ is the height of the prism $A B C A 1 B 1 C 1$, we have

$$
V_{A B C A} 1 B 1 C 1=S_{\triangle A B C} \cdot A K=\frac{a^{2} \sqrt{3}}{4} \cdot \frac{a \sqrt{3}}{2}=\frac{3 a^{3}}{8}
$$

## Answer  $\frac{3 a^{3}}{8}$