In a rectangular parallelepiped, one of the sides of the base is 8 cm. The diagonal of the parallelepiped is 16 cm
| August 5, 2021 | education
In a rectangular parallelepiped, one of the sides of the base is 8 cm. The diagonal of the parallelepiped is 16 cm and makes an angle of 45 ° with the side face containing this side. Find the volume of the parallelepiped.
In a right-angled triangle BB1D, one of the acute angles is 45, then triangle BB1D is isosceles, BB1 = BD.
Determine the lengths of the legs BB1 and BD.
Sin45 = BD / DB1.
ВD = DB1 * Sin450 = 16 * √2 / 2 = 8 * √2 cm.
BB1 = BD = 8 * √2 cm.
In a right-angled triangle ABD, we determine the length of the leg AB.
AB ^ 2 = BD ^ 2 – AD ^ 2 = 128 – 64 = 64.
AB = 8 cm.
Let’s define the volume of the parallelepiped.
V = AB * AD * BB1 = 8 * 8 * 8 * √2 = 512 * √2 cm3.
Answer: The volume of a parallelepiped is 512 * √2 cm3.
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