The base of a rectangular parallelepiped is a rhombus, one of whose angles is 30 degrees.

| September 17, 2021 | education

The base of a rectangular parallelepiped is a rhombus, one of whose angles is 30 degrees. Calculate the area of the diagonal section of the parallelepiped by the plane containing the smaller diagonal of the base if the volume of the parallelepiped is 18 cm3 and the area of its lateral surface is 48 cm3.

The volume of the parallelepiped is:

V = Sosn * AA1.

The lateral surface area of ​​the parallelepiped is:

Sside = Ravsd * AA1.

Let the side of the rhombus at the base of the parallelepiped be X cm.

Then Sosn = X ^ 2 * Sin30 = X ^ 2/2.

Then V = (X ^ 2/2) * AA1 = 18 cm3.

AA1 = 36 / X ^ 2 cm. (1)

P = 4 * X cm.

Then S side = 4 * X * AA1 = 48 cm2.

AA1 = 48 / (4 * X) = 12 / X. (2).

Let’s equate equalities 1 and 2.

36 / X ^ 2 = 12 / X.

12 * X ^ 2 = 36 * X.

X = AB = 3 cm.

Then AA1 = 36/9 = 4 cm.

By the cosine theorem, we determine the length of the diagonal BD.

ВD ^ 2 = 9 + 9 – 2 * 3 * 3 * √3 / 2 = 18 – 9 * √3 = 9 * (2 – √3) cm.

ВD = 3 * √ (2 – √3) see.

The diagonal section BB1D1D is a rectangle, then S section = BD * AA1 = 3 * √ (2 – √3) * 4 = 12 * √ (2 – √3) cm2.

Answer: The area of ​​the diagonal section is 12 * √ (2 – √3) cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.