The sides of the base of a straight parallelepiped are 7 cm and 3 roots of 2 cm, and the acute angle of the base is 45 degrees.

The sides of the base of a straight parallelepiped are 7 cm and 3 roots of 2 cm, and the acute angle of the base is 45 degrees. The smaller diagonal of the parallelepiped makes an angle of 45 degrees with the plane of the base. Find the volume of the parallelepiped.

At the base of the parallelepiped there is a parallelogram, the side edge is perpendicular to the plane of the base.
BD is the smaller diagonal since ∠A = 45 ° and ∠B = 135 °.
Therefore BD <AC.
ΔBB1D – rectangular, since BB1 = BD.
By the cosine theorem from triangle ABD:
BD ^ 2 = 49 + 18-2 * 7 * 3√2 * 1 / √2 = 49 + 18-42 = 7 + 18 = 25
BD = 5 cm.
S main-S (ABCD) = 7 * 3√2 * sin45 ° = 7 * 7√2 / √2 = 21 cm ^ 2
BD = BB1 = 5cm
Vpair = Sbn * BB1 = 5 * 21 = 105 cm ^ 3