[ Geometry (other)]

Find the volume of an oblique triangular prism, the base of which is an equilateral triangle with a side equal to $a$, if the lateral edge of the prism is equal to the side of the base and is inclined to the plane of the base 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 $A B C, A B$ $=B C=A C=A A_{1}=B B_{1}=D D_{1}=a$, According to the problem, $\angle A A_{1} K=60^{\circ}$

From the right triangle $A K A_{1}$, we find that

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

Since $A K$ is the height of the prism $A B C A_{1} B_{1} C_{1}$, then

$$
V\left(A B C A_{1} B_{1} C_{1}\right)=S(A B C) \cdot A K=\left(a^{2} \sqrt{3} / 4\right) \cdot(a \sqrt{3} / 2)=3 a^{3} / 8
$$

## Answer

$$
3 a^{3} / 8
$$

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