The sides of the bases and the diagonal of the rectangular parallelepiped are 1: 2: 3.

The sides of the bases and the diagonal of the rectangular parallelepiped are 1: 2: 3. The length of the side edge is 4. Find the volume of the parallelepiped.

Let the side AB = CD = X cm, then BC = AD = 2 * X cm, AC1 = 3 * X cm.

Let’s construct the AC diagonal. By the Pythagorean theorem, in a right-angled triangle ACD, AC ^ 2 = AD ^ 2 + CD ^ 2 = 4 * X ^ 2 + X ^ 2 = 5 * X ^ 2.

In a right-angled triangle ACC1, according to the Pythagorean theorem, CC1 ^ 2 = AC1 ^ 2 – AC ^ 2.

16 = 9 * X ^ 2 – 5 * X ^ 2 = 4 * X ^ 2.

X ^ 2 = 4.

X = 2.

Then AB = 2 cm, AD = 4 cm.

V = AB * AD * AA1 = 2 * 4 * 4 = 32 cm3.

Answer: The volume of a parallelepiped is 32 cm3.