The sides of the base of a straight parallelepiped are 4 √2 and 8 cm, the angle between them is 45 degrees.

The sides of the base of a straight parallelepiped are 4 √2 and 8 cm, the angle between them is 45 degrees. Find the volume of a straight parallelepiped if the smaller diagonal is 4√3.

base diagonal d = √ [(4√2) ^ 2 + 8 ^ 2-2 * 4√2 * 8 * cos45] = 4√2 cm
height h = √ [D ^ 2-d ^ 2] = √ [(4√3) ^ 2- (4√2) ^ 2] = 4 cm
base area S = 4√2 * 8 * sin45 = 32 cm2
volume V = S * h = 32 * 4 = 128 cm3