At the base of a rectangular prism is a rhombus, the diagonals of the prism make angles of 30 and 60 degrees

At the base of a rectangular prism is a rhombus, the diagonals of the prism make angles of 30 and 60 degrees with the base plane, the height of the prism = 6 cm. Find the volume of the prism.

Since, by condition, the prism is straight, the triangles AA1C and DD1B are rectangular.

In a right-angled triangle АА1С tg30 = АА1 / АС, then АС = АА1 / tg30 = 6 / (1 / √3) = 6 * √3 cm.

In a right-angled triangle DD1B, tg60 = DD1 / BD, then BD = DD1 / tg60 = 6 / √3 cm.

Let us determine the area of the base of the prism, which is the rhombus.

Sbn = АС * ВD / 2 = (6 * √3 * 6 / √3) / 2 = 18 cm2.

Let’s define the volume of the prism.

V = Sbn * AA1 = 18 * 6 = 108 cm3.

Answer: The volume of the prism is 108 cm3.