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

| September 17, 2021 | education

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 of the parallelepiped?

Determine the area of the base of the parallelepiped.

Since there is a square at the base, then Sosn = AD ^ 2 = a ^ 2 cm2.

The side faces of the parallelepiped are perpendicular, then the triangle A1B1D is rectangular with an acute angle A1DB1 = 30, then DV1 = 2 * A1B1 = 2 * a cm.

Then A1D ^ 2 = DB1 ^ 2 – A1B1 ^ 2 = 4 * a ^ 2 – a ^ 2 = 3 * a ^ 2.

A1D = a * √3 cm.

Triangle AA1D is rectangular, then AA1 ^ 2 = A1D ^ 2 – AD ^ 2 = 3 * a ^ 2 – a ^ 2 = 2 * a ^ 2.

AA1 = a * √2 cm.

Then V = Sax * AA1 = a ^ 2 * a * √2 = a ^ 3 * √2 cm3.

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



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