Find the volume of a parallelepiped, at the base of which is a parallelogram, with sides 2 and √3

Find the volume of a parallelepiped, at the base of which is a parallelogram, with sides 2 and √3, and an angle between them of 30 degrees, if the height of the pyramid is equal to the smaller diagonal of the base.

The volume of a parallelepiped is equal to the product of the area of ​​the base and the height:

V = S main * h.

The area of ​​the base is equal to the area of ​​a parallelogram with sides equal to 2 and √3, and an angle between them, which is 30 °. The area of ​​a parallelogram is defined as the product of the lengths of two adjacent sides hf sine of the angle between them:

Sbas = a * b * sin 30 ° = 2 * √3 * 0.5 = √3.

Obviously, in a parallelogram, the smaller diagonal lies opposite the smaller angle, this diagonal can be found by the cosine theorem:

d2 = a2 + b2 – 2 * a * b * cos 30 ° = 4 + 3 – 2 * 2 * √3 * √3 / 2 = 7 – 6 = 1;

d = 1 – the smaller diagonal of the base, equal to the height of the parallelepiped.

V = Sbase * h = √3 * 1 = √3 – the required volume.