In a regular hexagonal prism, all edges are 48. Find the distance between points D and B1.

Let’s construct a short diagonal of the ВD of a regular hexagon. In a regular hexagon, the angles at its vertices are 120, then the ВСD angle = 120.

In the triangle of the ВСD, by the cosine theorem, we determine the length of the side of the ВD.

ВD ^ 2 = СВ ^ 2 + СD ^ 2 – СВ * СD * Cos120 = 2304 + 2304 – 2 * 2304 * (-1/2) = 6912.

ВD = 48 * √3 cm.

The triangle DBВ1 is rectangular, with a right angle at point B, then, according to the Pythagorean theorem, DB1 ^ 2 = ВD ^ 2 + BB1 ^ 2 = 6912 + 2304 = 9216.

ВD = 96 cm.

Answer: the distance between points D and B1 is 96 cm.