The axial section of the cylinder is a square, the diagonal length of which is 36 cm. Find the radius of the base of the cylinder.

The axial section of a cylinder is a rectangle, one side of which is the height of the cylinder, the other is the diameter of its base. If the axial section of the cylinder is a square, then the diameter of the base is equal to the height of the cylinder.
Consider a right-angled triangle formed by the diameter of the base, the height of the cylinder and the diagonal of the axial section. The sum of the squares of the legs is equal to the square of the hypotenuse, which means we can write:
h ^ 2 + d ^ 2 = D ^ 2;
2d ^ 2 = D ^ 2;
2d ^ 2 = 36 ^ 2 = 1296;
d ^ 2 = 648;
d = 18√2 cm.
The diameter of the base of the cylinder is 18√2 cm, therefore, the radius of the base of this cylinder is 9√2 cm.