The diagonals of a straight parallelepiped are 8 and 10 cm, and the sides of the base are 5 and 3 cm. Find the volume

| March 1, 2021 | education

Let the angle BAD = α and α <90, then the angle ABC = (180 – α).

By the cosine theorem, we determine the lengths of the diagonals AC and BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cosα = 34 – 30 * Cosα.

AC ^ 2 = AB ^ 2 + BC – 2 * AB * BC * Cos (180 – α) = 34 + 30 * Cosα.

In triangles ACC1 and DBB1, according to the Pythagorean theorem, we express the legs BB1 and CC1.

BB1 ^ 2 = DB1 ^ 2 – BD ^ 2 = 64 – 34 +30 * Cosα = 30 + 30 * Cosα.

CC1 ^ 2 = AC1 ^ 2 – AC ^ 2 = 100 – 34 – 30 * Cosα = 66 – 30 * Cosα.

Since BB1 ​​^ 2 = CC1 ^ 2, then 30 + 30 * Cosα = 66 – 30 * Cosα.

60 * Cosα = 36.

Cosα = 36/60 = 0.6.

Sinα = √ (1 – 0.36) = 0.8.

Then Sbn = AB * AD * Sinα = 3 * 5 * 0.8 = 12 cm2.

BB1 ^ 2 = 30 + 30 * 0.6 = 48.

BB1 = 4 * √3 cm.

Then V = BB1 * Sbn = 4 * √3 * 12 = 48 * √3 cm3.

Answer: The volume of a parallelepiped is 48 * √3 cm3.



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.