2.18. A regular hexagonal prism, whose lateral edges are equal to $3 \mathrm{~cm}$, is cut by a diagonal plane into two equal quadrilateral prisms. Determine the volume of the hexagonal prism if the lateral surface area of the quadrilateral prism is 30 cm $^{2}$.
2.18. Let $AB = a$ (Fig. 2.17); then the desired volume $V = 6 \cdot \frac{a^2 \sqrt{3}}{4} \cdot h$, where $h = 3$ cm is the height of the prism. It remains to find $a$. According to the problem, the lateral surface area of the prism $ABCD A_1 B_1 C_1 D_1$ is $30 \, \text{cm}^2$. But $S_{\text{lat}} = (3 AB + AD) h$; here $AD = 2a$ (as the diameter of the circle circumscribed around the regular hexagon). Therefore, $5a \cdot 3 = 30$, from which $a = 2 \, \text{cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-464.jpg?height=467&width=419&top_left_y=1316&top_left_x=733)

Fig. 2.17

Finally, we have

$$
V = 6 \cdot \frac{2^2 \sqrt{3}}{4} \cdot 3 = 18 \sqrt{3} \, \text{cm}^3
$$

Answer: $18 \sqrt{3} \, \text{cm}^3$.