A regular hexagonal prism has two body diagonals of lengths 12 and 13. What is the volume of the prism?
The two types of body diagonals of the prism can be determined from the two planar sections of the prism. First, we fit a plane perpendicular to the base through the longest diagonal of the hexagon, and then through the shorter diagonal. (We have no other choice, as the prism has only two different lengths of body diagonals.) The intersection in both cases will be a rectangle. One side of the rectangle will be equal to the height of the prism. The other side of the rectangle will be $2a$ in the first case and $a \sqrt{3}$ in the second case, where $a$ is the length of the sides of the regular hexagon.

The lengths of the diagonals can be written using the Pythagorean theorem as:

$$
13^{2}=(2 a)^{2}+m^{2}
$$

and

$$
12^{2}=(a \sqrt{3})^{2}+m^{2}
$$

From the system of equations, we get $a=5, m=\sqrt{69}$.

The volume of the prism is: $V=A_{t} \cdot m$, where $A_{t}=6 \frac{a^{2} \sqrt{3}}{4}$, that is,

$$
V=6 \cdot \frac{25 \sqrt{3}}{4} \sqrt{69} \approx 539.53 \text { cubic units. }
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_c0bcf1e0f29d4e5a28c2g-1.jpg?height=994&width=314&top_left_y=972&top_left_x=900)