What is the volume of a rectangular parallelepiped, the diagonals of whose faces are equal: √5, √10, √13?

Let us denote the lengths of the sides of the parallelepiped through X, Y, Z.

AD= BC = A1D1 = B1C1 = X cm.

AB = SD = A1B1 = C1D1 = Y cm.

AA1 = BB1 = CC1 = DD1 = Z see.

Then:

A1D ^ 2 = 13 = X ^ 2 + Z ^ 2.

ВD ^ 2 = 10 = X ^ 2 + Y ^ 2.

DС1 ^ 2 = 5 = Y ^ 2 + Z ^ 2.

Let us solve the system of three equations by the substitution method.

X ^ 2 = 13 – Z ^ 2.

Y ^ 2 = 5 – Z ^ 2.

10 = 13 – Z ^ 2 + 5 – Z ^ 2.

2 * Z ^ 2 = 18 – 10 = 8.

Z ^ 2 = 4.

Z = 2 cm.

X ^ 2 = 13 – Z ^ 2 = 13 – 4 = 9.

X = 3 cm.

Y ^ 2 = 5 – Z2 = 5 – 4 = 1.

Y = 1 cm.

Let’s define the volume of the parallelepipeds:

V = X * Y * X = 3 * 1 * 2 = 6 cm3.

Answer: The volume of a parallelepiped is 6 cm3.