The axial section of the cylinder is a square with a diagonal of 12 cm, calculate the volume of the cylinder.

Since the axial section of AВСD is square, AB = BC, and triangle ABC is rectangular and isosceles.
Then, by the Pythagorean theorem, AC ^ 2 = 2 * AB ^ 2 = 2 * BC ^ 2.
AB ^ 2 = BC ^ 2 = AC ^ 2/2 = 144/2 = 72.
AB = BC = 6 * √2 cm.
Determine the area of the base of the cylinder.
Sb = π * AD ^ 2/4 = π * 72/4 = 18 * π cm2.
Then the volume of the cylinder is: Vcyl = Sosn * AB = 18 * π * 6 * √2 = 108 * π * √2 cm3.
Answer: The volume of the cylinder is 108 * π * √2 cm3.