The base of the rectangular parallelepiped is a square with side a. the diagonal of the parallelepiped makes

The base of the rectangular parallelepiped is a square with side a. the diagonal of the parallelepiped makes an angle of 30 degrees with the side face. find the volume.

Since the parallelepiped is straight, the triangle DAB1 is rectangular, in which tg30 = AD / AB1.

AB1 = AD / tg30 = a / (√3 / 3) = a * √3 cm.

In a right-angled triangle ABB1, according to the Pythagorean theorem, BB1 ^ 2 = AB1 ^ 2 – AB ^ 2 = 3 * a ^ 2 – a ^ 2 = 2 * a ^ 2.

BB1 = a * √2 cm.

Determine the area of the base of the parallelepiped. Sosn = AB * AD = a * a = a ^ 2.

Then V = Sax * BB1 = a ^ 2 * a * √2 = a ^ 3 * √2 cm3.

Answer: The volume of a parallelepiped is a ^ 3 * √2 cm3.